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מדריכי לימוד > Prealgebra

Problem Set 11: Geometry

Practice Makes Perfect

Use the Properties of Angles In the following exercises, find ⓐ the supplement and ⓑ the complement of the given angle. \text{53^\circ } ⓐ 127° ⓑ 37° \text{16^\circ } \text{29^\circ } ⓐ 151° ⓑ 61° \text{72^\circ } In the following exercises, use the properties of angles to solve. Find the supplement of a \text{135^\circ } angle. 45° Find the complement of a \text{38^\circ } angle. Find the complement of a 27.5∘27.5^\circ angle. 62.5° Find the supplement of a 109.5∘109.5^\circ angle. Two angles are supplementary. The larger angle is \text{56^\circ } more than the smaller angle. Find the measures of both angles. 62°, 118° Two angles are supplementary. The smaller angle is \text{36^\circ } less than the larger angle. Find the measures of both angles. Two angles are complementary. The smaller angle is \text{34^\circ } less than the larger angle. Find the measures of both angles. 62°, 28° Two angles are complementary. The larger angle is \text{52^\circ } more than the smaller angle. Find the measures of both angles. Use the Properties of Triangles In the following exercises, solve using properties of triangles. The measures of two angles of a triangle are \text{26^\circ } and \text{98^\circ }. Find the measure of the third angle. 56° The measures of two angles of a triangle are \text{61^\circ } and \text{84^\circ }. Find the measure of the third angle. The measures of two angles of a triangle are \text{105^\circ } and \text{31^\circ }. Find the measure of the third angle. 44° The measures of two angles of a triangle are \text{47^\circ } and \text{72^\circ }. Find the measure of the third angle. One angle of a right triangle measures \text{33^\circ }. What is the measure of the other angle? 57° One angle of a right triangle measures \text{51^\circ }. What is the measure of the other angle? One angle of a right triangle measures 22.5∘22.5^\circ . What is the measure of the other angle? 67.5° One angle of a right triangle measures 36.5∘36.5^\circ . What is the measure of the other angle? The two smaller angles of a right triangle have equal measures. Find the measures of all three angles. 45°, 45°, 90° The measure of the smallest angle of a right triangle is \text{20^\circ } less than the measure of the other small angle. Find the measures of all three angles. The angles in a triangle are such that the measure of one angle is twice the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles. 30°, 60°, 90° The angles in a triangle are such that the measure of one angle is \text{20^\circ } more than the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles. Find the Length of the Missing Side In the following exercises, ΔABC\Delta ABC is similar to ΔXYZ\Delta XYZ. Find the length of the indicated side. Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled 9, the side across from B is labeled b, and the side across from C is labeled 15. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled x, the side across from Y is labeled 8, and the side across from Z is labeled 10. side bb 12 side xx On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides are shown in the figure below. The actual distance from Los Angeles to Las Vegas is 270270 miles. A triangle is shown. The vertices are labeled San Francisco, Las Vegas, and Los Angeles. The side across from San Francisco is labeled 1 inch, the side across from Las Vegas is labeled 1.3 inches, and the side across from Los Angeles is labeled 2.1 inches. Find the distance from Los Angeles to San Francisco. 351 miles Find the distance from San Francisco to Las Vegas. Use the Pythagorean Theorem In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse. A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 9, the other as 12. 15 A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 16, the other as 12. A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 15, the other as 20. 25 A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 5, the other as 12. Find the Length of the Missing Side In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary. A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 10. One of the sides touching the right angle is labeled as 6. 8 A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 8. A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 5. 12 A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 20. One of the sides touching the right angle is labeled as 16. A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 8. 10.2 A right triangle is shown. The right angle is marked with a box. Both of the sides touching the right angle are labeled as 6. A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15. 8 A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 7. One of the sides touching the right angle is labeled as 5. In the following exercises, solve. Approximate to the nearest tenth, if necessary. A 13-foot\text{13-foot} string of lights will be attached to the top of a 12-foot\text{12-foot} pole for a holiday display. How far from the base of the pole should the end of the string of lights be anchored? A vertical pole is shown with a string of lights going from the top of the pole to the ground. The pole is labeled 12 feet. The string of lights is labeled 13 feet. 5 feet Pam wants to put a banner across her garage door to congratulate her son on his college graduation. The garage door is 1212 feet high and 1616 feet wide. How long should the banner be to fit the garage door? A picture of a house is shown. The rectangular garage is 12 feet high and 16 feet wide. A blue banner goes diagonally across the garage. Chi is planning to put a path of paving stones through her flower garden. The flower garden is a square with sides of 1010 feet. What will the length of the path be? A square garden is shown. One side is labeled as 10 feet. There is a diagonal path of blue circular stones going from the lower left corner to the upper right corner. 14.1 feet Brian borrowed a 20-foot\text{20-foot} extension ladder to paint his house. If he sets the base of the ladder 66 feet from the house, how far up will the top of the ladder reach? A picture of a house is shown with a ladder leaning against it. The ladder is labeled 20 feet tall. The horizontal distance from the house to the base of the ladder is 6 feet.

Everyday Math

Building a scale model Joe wants to build a doll house for his daughter. He wants the doll house to look just like his house. His house is 3030 feet wide and 3535 feet tall at the highest point of the roof. If the dollhouse will be 2.52.5 feet wide, how tall will its highest point be? 2.9 feet Measurement A city engineer plans to build a footbridge across a lake from point X\text{X} to point Y\text{Y}, as shown in the picture below. To find the length of the footbridge, she draws a right triangle XYZ\text{XYZ}, with right angle at X\text{X}. She measures the distance from X\text{X} to Z,800\text{Z},800 feet, and from Y\text{Y} to Z,1,000\text{Z},1,000 feet. How long will the bridge be? A lake is shown. Point Y is on one side of the lake, directly across from point X. Point Z is on the same side of the lake as point X.

Writing Exercises

Write three of the properties of triangles from this section and then explain each in your own words. Answers will vary. Explain how the figure below illustrates the Pythagorean Theorem for a triangle with legs of length 33 and 44. Three squares are shown, forming a right triangle in the center. Each square is divided into smaller squares. The smallest square is divided into 9 small squares. The medium square is divided into 16 small squares. The large square is divided into 25 small squares.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. . ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Use Properties of Angles, Triangles, and the Pythagorean Theorem

Use Properties of Angles In the following exercises, solve using properties of angles. What is the supplement of a \text{48^\circ } angle? 132° What is the complement of a \text{61^\circ } angle? Two angles are complementary. The smaller angle is \text{24^\circ } less than the larger angle. Find the measures of both angles. 33°, 57° Two angles are supplementary. The larger angle is \text{45^\circ } more than the smaller angle. Find the measures of both angles. Use Properties of Triangles In the following exercises, solve using properties of triangles. The measures of two angles of a triangle are 2222 and 8585 degrees. Find the measure of the third angle. 73° One angle of a right triangle measures 41.541.5 degrees. What is the measure of the other small angle? One angle of a triangle is \text{30^\circ } more than the smallest angle. The largest angle is the sum of the other angles. Find the measures of all three angles. 30°, 60°, 90° One angle of a triangle is twice the measure of the smallest angle. The third angle is \text{60^\circ } more than the measure of the smallest angle. Find the measures of all three angles. In the following exercises, ΔABC\Delta ABC is similar to ΔXYZ\Delta XYZ. Find the length of the indicated side. Two triangles are shown. Triangle ABC is on the left. The side across from A is labeled 21, across from B is b, and across from C is 11.2. Triangle XYZ is on the right. The side across from X is labeled x, across from Y is 10, and across from Z is 8. side xx 15 side bb Use the Pythagorean Theorem In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary. A right triangle is shown. The base is labeled 10, the height is labeled 24. 26 A right triangle is shown. The base is labeled 6, the height is labeled 8. A right triangle is shown. The height is labeled 15, the hypotenuse is labeled 17. 8 A right triangle is shown. The height is labeled 15, the hypotenuse is labeled 25. A right triangle is shown. The height is labeled 7, the base is labeled 4. 8.1 A right triangle is shown. The height is labeled 11, the base is labeled 10. In the following exercises, solve. Approximate to the nearest tenth, if necessary. Sergio needs to attach a wire to hold the antenna to the roof of his house, as shown in the figure. The antenna is 88 feet tall and Sergio has 1010 feet of wire. How far from the base of the antenna can he attach the wire? An image of a house is shown. A 10-foot wire is going from the roof of the house to the ground. The wire hits the house at a height of 8 feet. 6 feet Seong is building shelving in his garage. The shelves are 3636 inches wide and 1515 inches tall. He wants to put a diagonal brace across the back to stabilize the shelves, as shown. How long should the brace be? A rectangular shelf is shown, with a diagonal drawn in from the lower left corner to the upper right corner. The side is labeled 15 inches, the top is labeled 36 inches.

Use Properties of Rectangles, Triangles, and Trapezoids

Understand Linear, Square, Cubic Measure In the following exercises, would you measure each item using linear, square, or cubic measure? amount of sand in a sandbag cubic height of a tree size of a patio square length of a highway In the following exercises, find ⓐ the perimeter ⓑ the area of each figure Three squares are shown, in a sideways L shape. ⓐ 8 units ⓑ 3 sq. units Five squares are shown, in a T-shape. There are three squares across the top and three squares down. Use Properties of Rectangles In the following exercises, find the ⓐ perimeter ⓑ area of each rectangle The length of a rectangle is 4242 meters and the width is 2828 meters. ⓐ 140 m ⓑ 1176 sq. m The length of a rectangle is 3636 feet and the width is 1919 feet. A sidewalk in front of Kathy’s house is in the shape of a rectangle 44 feet wide by 4545 feet long. ⓐ 98 ft. ⓑ 180 sq. ft. A rectangular room is 1616 feet wide by 1212 feet long. In the following exercises, solve. Find the length of a rectangle with perimeter of 220220 centimeters and width of 8585 centimeters. 25 cm Find the width of a rectangle with perimeter 3939 and length 1111. The area of a rectangle is 23562356 square meters. The length is 3838 meters. What is the width? 62 m The width of a rectangle is 4545 centimeters. The area is 27002700 square centimeters. What is the length? The length of a rectangle is 1212 centimeters more than the width. The perimeter is 7474 centimeters. Find the length and the width. 24.5 in., 12.5 in. The width of a rectangle is 33 more than twice the length. The perimeter is 9696 inches. Find the length and the width. Use Properties of Triangles In the following exercises, solve using the properties of triangles. Find the area of a triangle with base 1818 inches and height 1515 inches. 135 sq. in. Find the area of a triangle with base 3333 centimeters and height 2121 centimeters. A triangular road sign has base 3030 inches and height 4040 inches. What is its area? 600 sq. in. If a triangular courtyard has sides 99 feet and 1212 feet and the perimeter is 3232 feet, how long is the third side? A tile in the shape of an isosceles triangle has a base of 66 inches. If the perimeter is 2020 inches, find the length of each of the other sides. 7 in., 7 in. Find the length of each side of an equilateral triangle with perimeter of 8181 yards. The perimeter of a triangle is 5959 feet. One side of the triangle is 33 feet longer than the shortest side. The third side is 55 feet longer than the shortest side. Find the length of each side. 17 ft., 20 ft., 22 ft. One side of a triangle is three times the smallest side. The third side is 99 feet more than the shortest side. The perimeter is 3939 feet. Find the lengths of all three sides. Use Properties of Trapezoids In the following exercises, solve using the properties of trapezoids. The height of a trapezoid is 88 feet and the bases are 1111 and 1414 feet. What is the area? 100 sq. ft. The height of a trapezoid is 55 yards and the bases are 77 and 1010 yards. What is the area? Find the area of the trapezoid with height 2525 meters and bases 32.532.5 and 21.521.5 meters. 675 sq. m A flag is shaped like a trapezoid with height 6262 centimeters and the bases are 91.591.5 and 78.178.1 centimeters. What is the area of the flag?

Practice Makes Perfect

Understand Linear, Square, and Cubic Measure In the following exercises, determine whether you would measure each item using linear, square, or cubic units. amount of water in a fish tank cubic length of dental floss living area of an apartment square floor space of a bathroom tile height of a doorway linear capacity of a truck trailer In the following exercises, find the ⓐ perimeter and ⓑ area of each figure. Assume each side of the square is 11 cm. A rectangle is shown comprised of 4 squares forming a horizontal line. ⓐ 10 cm ⓑ 4 sq. cm A rectangle is shown comprised of 3 squares forming a vertical line. Three squares are shown. There is one on the bottom left, one on the bottom right, and one on the top right. ⓐ 8 cm ⓑ 3 sq. cm Four squares are shown. Three form a horizontal line, and there is one above the center square. Five squares are shown. There are three forming a horizontal line across the top and two underneath the two on the right. ⓐ 10 cm ⓑ 5 sq. cm A square is shown. It is comprised of nine smaller squares. Use the Properties of Rectangles In the following exercises, find the ⓐ perimeter and ⓑ area of each rectangle. The length of a rectangle is 8585 feet and the width is 4545 feet. ⓐ 260 ft ⓑ 3825 sq. ft The length of a rectangle is 2626 inches and the width is 5858 inches. A rectangular room is 1515 feet wide by 1414 feet long. ⓐ 58 ft ⓑ 210 sq. ft A driveway is in the shape of a rectangle 2020 feet wide by 3535 feet long. In the following exercises, solve. Find the length of a rectangle with perimeter 124124 inches and width 3838 inches. 24 inches Find the length of a rectangle with perimeter 20.220.2 yards and width of 7.87.8 yards. Find the width of a rectangle with perimeter 9292 meters and length 1919 meters. 27 meters Find the width of a rectangle with perimeter 16.216.2 meters and length 3.23.2 meters. The area of a rectangle is 414414 square meters. The length is 1818 meters. What is the width? 23 m The area of a rectangle is 782782 square centimeters. The width is 1717 centimeters. What is the length? The length of a rectangle is 99 inches more than the width. The perimeter is 4646 inches. Find the length and the width. 7 in., 16 in. The width of a rectangle is 88 inches more than the length. The perimeter is 5252 inches. Find the length and the width. The perimeter of a rectangle is 5858 meters. The width of the rectangle is 55 meters less than the length. Find the length and the width of the rectangle. 17 m, 12 m The perimeter of a rectangle is 6262 feet. The width is 77 feet less than the length. Find the length and the width. The width of the rectangle is 0.70.7 meters less than the length. The perimeter of a rectangle is 52.652.6 meters. Find the dimensions of the rectangle. 13.5 m, 12.8 m The length of the rectangle is 1.11.1 meters less than the width. The perimeter of a rectangle is 49.449.4 meters. Find the dimensions of the rectangle. The perimeter of a rectangle of 150150 feet. The length of the rectangle is twice the width. Find the length and width of the rectangle. 25 ft, 50 ft The length of a rectangle is three times the width. The perimeter is 7272 feet. Find the length and width of the rectangle. The length of a rectangle is 33 meters less than twice the width. The perimeter is 3636 meters. Find the length and width. 7 m, 11 m The length of a rectangle is 55 inches more than twice the width. The perimeter is 3434 inches. Find the length and width. The width of a rectangular window is 2424 inches. The area is 624624 square inches. What is the length? 26 in. The length of a rectangular poster is 2828 inches. The area is 13161316 square inches. What is the width? The area of a rectangular roof is 23102310 square meters. The length is 4242 meters. What is the width? 55 m The area of a rectangular tarp is 132132 square feet. The width is 1212 feet. What is the length? The perimeter of a rectangular courtyard is 160160 feet. The length is 1010 feet more than the width. Find the length and the width. 35 ft, 45 ft The perimeter of a rectangular painting is 306306 centimeters. The length is 1717 centimeters more than the width. Find the length and the width. The width of a rectangular window is 4040 inches less than the height. The perimeter of the doorway is 224224 inches. Find the length and the width. 76 in., 36 in. The width of a rectangular playground is 77 meters less than the length. The perimeter of the playground is 4646 meters. Find the length and the width. Use the Properties of Triangles In the following exercises, solve using the properties of triangles. Find the area of a triangle with base 1212 inches and height 55 inches. 60 sq. in. Find the area of a triangle with base 4545 centimeters and height 3030 centimeters. Find the area of a triangle with base 8.38.3 meters and height 6.16.1 meters. 25.315 sq. m Find the area of a triangle with base 24.224.2 feet and height 20.520.5 feet. A triangular flag has base of 11 foot and height of 1.51.5 feet. What is its area? 0.75 sq. ft A triangular window has base of 88 feet and height of 66 feet. What is its area? If a triangle has sides of 66 feet and 99 feet and the perimeter is 2323 feet, how long is the third side? 8 ft If a triangle has sides of 1414 centimeters and 1818 centimeters and the perimeter is 4949 centimeters, how long is the third side? What is the base of a triangle with an area of 207207 square inches and height of 1818 inches? 23 in. What is the height of a triangle with an area of 893893 square inches and base of 3838 inches? The perimeter of a triangular reflecting pool is 3636 yards. The lengths of two sides are 1010 yards and 1515 yards. How long is the third side? 11 ft A triangular courtyard has perimeter of 120120 meters. The lengths of two sides are 3030 meters and 5050 meters. How long is the third side? An isosceles triangle has a base of 2020 centimeters. If the perimeter is 7676 centimeters, find the length of each of the other sides. 28 cm An isosceles triangle has a base of 2525 inches. If the perimeter is 9595 inches, find the length of each of the other sides. Find the length of each side of an equilateral triangle with a perimeter of 5151 yards. 17 ft Find the length of each side of an equilateral triangle with a perimeter of 5454 meters. The perimeter of an equilateral triangle is 1818 meters. Find the length of each side. 6 m The perimeter of an equilateral triangle is 4242 miles. Find the length of each side. The perimeter of an isosceles triangle is 4242 feet. The length of the shortest side is 1212 feet. Find the length of the other two sides. 15 ft The perimeter of an isosceles triangle is 8383 inches. The length of the shortest side is 2424 inches. Find the length of the other two sides. A dish is in the shape of an equilateral triangle. Each side is 88 inches long. Find the perimeter. 24 in. A floor tile is in the shape of an equilateral triangle. Each side is 1.51.5 feet long. Find the perimeter. A road sign in the shape of an isosceles triangle has a base of 3636 inches. If the perimeter is 9191 inches, find the length of each of the other sides. 27.5 in. A scarf in the shape of an isosceles triangle has a base of 0.750.75 meters. If the perimeter is 22 meters, find the length of each of the other sides. The perimeter of a triangle is 3939 feet. One side of the triangle is 11 foot longer than the second side. The third side is 22 feet longer than the second side. Find the length of each side. 12 ft, 13 ft, 14 ft The perimeter of a triangle is 3535 feet. One side of the triangle is 55 feet longer than the second side. The third side is 33 feet longer than the second side. Find the length of each side. One side of a triangle is twice the smallest side. The third side is 55 feet more than the shortest side. The perimeter is 1717 feet. Find the lengths of all three sides. 3 ft, 6 ft, 8 ft One side of a triangle is three times the smallest side. The third side is 33 feet more than the shortest side. The perimeter is 1313 feet. Find the lengths of all three sides. Use the Properties of Trapezoids In the following exercises, solve using the properties of trapezoids. The height of a trapezoid is 1212 feet and the bases are 99 and 1515 feet. What is the area? 144 sq. ft The height of a trapezoid is 2424 yards and the bases are 1818 and 3030 yards. What is the area? Find the area of a trapezoid with a height of 5151 meters and bases of 4343 and 6767 meters. 2805 sq. m Find the area of a trapezoid with a height of 6262 inches and bases of 5858 and 7575 inches. The height of a trapezoid is 1515 centimeters and the bases are 12.512.5 and 18.318.3 centimeters. What is the area? 231 sq. cm The height of a trapezoid is 4848 feet and the bases are 38.638.6 and 60.260.2 feet. What is the area? Find the area of a trapezoid with a height of 4.24.2 meters and bases of 8.18.1 and 5.55.5 meters. 28.56 sq. m Find the area of a trapezoid with a height of 32.532.5 centimeters and bases of 54.654.6 and 41.441.4 centimeters. Laurel is making a banner shaped like a trapezoid. The height of the banner is 33 feet and the bases are 44 and 55 feet. What is the area of the banner? 13.5 sq. ft Niko wants to tile the floor of his bathroom. The floor is shaped like a trapezoid with width 55 feet and lengths 55 feet and 88 feet. What is the area of the floor? Theresa needs a new top for her kitchen counter. The counter is shaped like a trapezoid with width 18.518.5 inches and lengths 6262 and 5050 inches. What is the area of the counter? 1036 sq. in. Elena is knitting a scarf. The scarf will be shaped like a trapezoid with width 88 inches and lengths 48.248.2 inches and 56.256.2 inches. What is the area of the scarf?

Everyday Math

Fence Jose just removed the children’s playset from his back yard to make room for a rectangular garden. He wants to put a fence around the garden to keep out the dog. He has a 5050 foot roll of fence in his garage that he plans to use. To fit in the backyard, the width of the garden must be 1010 feet. How long can he make the other side if he wants to use the entire roll of fence? 15 ft Gardening Lupita wants to fence in her tomato garden. The garden is rectangular and the length is twice the width. It will take 4848 feet of fencing to enclose the garden. Find the length and width of her garden. Fence Christa wants to put a fence around her triangular flowerbed. The sides of the flowerbed are 66 feet, 88 feet, and 1010 feet. The fence costs $10 per foot. How much will it cost for Christa to fence in her flowerbed? $24 Painting Caleb wants to paint one wall of his attic. The wall is shaped like a trapezoid with height 88 feet and bases 2020 feet and 1212 feet. The cost of the painting one square foot of wall is about $0.05. About how much will it cost for Caleb to paint the attic wall? A right trapezoid is shown.

Writing Exercises

If you need to put tile on your kitchen floor, do you need to know the perimeter or the area of the kitchen? Explain your reasoning. Answers will vary. If you need to put a fence around your backyard, do you need to know the perimeter or the area of the backyard? Explain your reasoning. Look at the two figures. A rectangle is shown on the left. It is labeled as 2 by 8. A square is shown on the right. It is labeled as 4 by 4. ⓐ Which figure looks like it has the larger area? Which looks like it has the larger perimeter? ⓑ Now calculate the area and perimeter of each figure. Which has the larger area? Which has the larger perimeter? Answers will vary. The length of a rectangle is 55 feet more than the width. The area is 5050 square feet. Find the length and the width. ⓐ Write the equation you would use to solve the problem. ⓑ Why can’t you solve this equation with the methods you learned in the previous chapter?

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. . ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

Solve Geometry Applications: Circles and Irregular Figures

Use Properties of Circles In the following exercises, solve using the properties of circles. Round answers to the nearest hundredth. A circular mosaic has radius 33 meters. Find the ⓐ circumference ⓑ area of the mosaic ⓐ 18.84 m ⓑ 28.26 sq. m A circular fountain has radius 88 feet. Find the ⓐ circumference ⓑ area of the fountain Find the diameter of a circle with circumference 150.72150.72 inches. 48 in. Find the radius of a circle with circumference 345.4345.4 centimeters Find the Area of Irregular Figures In the following exercises, find the area of each shaded region. A geometric shape is shown, formed by two rectangles. The top is labeled 8. The width of the top rectangle is labeled 3. The right side of the figure is labeled 5. The width of the bottom rectangle is labeled 3. 30 sq. units A geometric shape is shown. It is a U-shape. The base is labeled 5, the height 6. The horizontal and vertical lines at the top are labeled 2. A geometric shape is shown. It is formed by two triangles. The shared base of the two triangles is labeled 20. The height of each triangle is labeled 15. 300 sq. units A geometric shape is shown. It is a trapezoid with a triangle attached to the top on the right side. The height of the trapezoid is labeled 8, the bottom base is labeled 12, and the top is labeled 9. The height of the triangle is labeled 8. A geometric shape is shown. It is a rectangle with a semi-circle attached to the top. The base of the rectangle, also the diameter of the semi-circle, is labeled 10. The height of the rectangle is labeled 16. 199.25 sq. units A geometric shape is shown. It is a triangle with a semicircle attached. The base of the triangle, also the diameter of the semi-circle, is labeled 5. The height of the triangle is also labeled 5.

Solve Geometry Applications: Volume and Surface Area

Find Volume and Surface Area of Rectangular Solids In the following exercises, find the ⓐ volume ⓑ surface area of the rectangular solid a rectangular solid with length 1414 centimeters, width 4.54.5 centimeters, and height 1010 centimeters ⓐ 630 cu. cm ⓑ 496 sq. cm a cube with sides that are 33 feet long a cube of tofu with sides 2.52.5 inches ⓐ 15.625 cu. in. ⓑ 37.5 sq. in. a rectangular carton with length 3232 inches, width 1818 inches, and height 1010 inches Find Volume and Surface Area of Spheres In the following exercises, find the ⓐ volume ⓑ surface area of the sphere. a sphere with radius 44 yards ⓐ 267.95 cu. yd. ⓑ 200.96 sq. yd. a sphere with radius 1212 meters a baseball with radius 1.451.45 inches ⓐ 12.76 cu. in. ⓑ 26.41 sq. in. a soccer ball with radius 2222 centimeters Find Volume and Surface Area of Cylinders In the following exercises, find the ⓐ volume ⓑ surface area of the cylinder a cylinder with radius 22 yards and height 66 yards ⓐ 75.36 cu. yd. ⓑ 100.48 sq. yd. a cylinder with diameter 1818 inches and height 4040 inches a juice can with diameter 88 centimeters and height 1515 centimeters ⓐ 753.6 cu. cm ⓑ 477.28 sq. cm a cylindrical pylon with diameter 0.80.8 feet and height 2.52.5 feet Find Volume of Cones In the following exercises, find the volume of the cone. a cone with height 55 meters and radius 11 meter 5.233 cu. m a cone with height 2424 feet and radius 88 feet a cone-shaped water cup with diameter 2.62.6 inches and height 2.62.6 inches 4.599 cu. in. a cone-shaped pile of gravel with diameter 66 yards and height 55 yards

Find the complement of a \text{52^\circ } angle. 38° The measure of one angle of a triangle is twice the measure of the smallest angle. The measure of the third angle is 1414 more than the measure of the smallest angle. Find the measures of all three angles. The perimeter of an equilateral triangle is 145145 feet. Find the length of each side. 48.3 ΔABC\Delta ABC is similar to ΔXYZ\Delta XYZ. Find the length of side cc. Two triangles are shown. Triangle XYZ is on the left. The side across from X is labeled 5, the side across from Y is labeled 10, the side across from Z is labeled 7. Triangle ABC is on the right. The side across from A is labeled 6, the side across from B is labeled 12, and the side across from C is labeled c. Find the length of the missing side. Round to the nearest tenth, if necessary. A right triangle is shown. The height is labeled 24 and the hypotenuse is labeled 26. 10 Find the length of the missing side. Round to the nearest tenth, if necessary. A right triangle is shown. The base is labeled 6 and the height is labeled 9. A baseball diamond is shaped like a square with sides 9090 feet long. How far is it from home plate to second base, as shown? A baseball diamond is shown. It is in the shape of a sideways square. The bottom corner is labeled Home and there is a dotted line to the top corner, labeled 2nd base. The right corner is labeled 1st base and the left corner is labeled 3rd base. 127.3 ft The length of a rectangle is 22 feet more than five times the width. The perimeter is 4040 feet. Find the dimensions of the rectangle. A triangular poster has base 8080 centimeters and height 5555 centimeters. Find the area of the poster. 2200 square centimeters A trapezoid has height 1414 inches and bases 2020 inches and 2323 inches. Find the area of the trapezoid. A circular pool has diameter 9090 inches. What is its circumference? Round to the nearest tenth. 282.6 inches Find the area of the shaded region. Round to the nearest tenth. A geometric shape is shown. It is a rectangle with a semi-circle attached on the left and a triangle attached on the right. The height of the rectangle, also the height of the triangle and the diameter of the semi-circle, is labeled 4. The base of the figure is labeled 10. The top of the rectangle is labeled 7. Find the volume of a rectangular room with width 1212 feet, length 1515 feet, and height 88 feet. 1440 A coffee can is shaped like a cylinder with height 77 inches and radius 55 inches. Find (a) the surface area and (b) the volume of the can. Round to the nearest tenth. A traffic cone has height 7575 centimeters. The radius of the base is 2020 centimeters. Find the volume of the cone. Round to the nearest tenth. 31,400 cubic inches Solve the formula A=12bhA=\frac{1}{2}bh for h:h\text{:} ⓐ when A=1716A=1716 and b=66b=66 ⓑ in general ⓐ height=52\text{height}=52 ⓑ h=2Abh=\frac{2A}{b} Solve x+5y=14x+5y=14 for yy.

Practice Makes Perfect

Use the Properties of Circles In the following exercises, solve using the properties of circles. The lid of a paint bucket is a circle with radius 77 inches. Find the ⓐ circumference and ⓑ area of the lid. ⓐ 43.96 in. ⓑ 153.86 sq. in. An extra-large pizza is a circle with radius 88 inches. Find the ⓐ circumference and ⓑ area of the pizza. A farm sprinkler spreads water in a circle with radius of 8.58.5 feet. Find the ⓐ circumference and ⓑ area of the watered circle. ⓐ 53.38 ft ⓑ 226.865 sq. ft A circular rug has radius of 3.53.5 feet. Find the ⓐ circumference and ⓑ area of the rug. A reflecting pool is in the shape of a circle with diameter of 2020 feet. What is the circumference of the pool? 62.8 ft A turntable is a circle with diameter of 1010 inches. What is the circumference of the turntable? A circular saw has a diameter of 1212 inches. What is the circumference of the saw? 37.68 in. A round coin has a diameter of 33 centimeters. What is the circumference of the coin? A barbecue grill is a circle with a diameter of 2.22.2 feet. What is the circumference of the grill? 6.908 ft The top of a pie tin is a circle with a diameter of 9.59.5 inches. What is the circumference of the top? A circle has a circumference of 163.28163.28 inches. Find the diameter. 52 in. A circle has a circumference of 59.6659.66 feet. Find the diameter. A circle has a circumference of 17.2717.27 meters. Find the diameter. 5.5 m A circle has a circumference of 80.0780.07 centimeters. Find the diameter. In the following exercises, find the radius of the circle with given circumference. A circle has a circumference of 150.72150.72 feet. 24 ft A circle has a circumference of 251.2251.2 centimeters. A circle has a circumference of 40.8240.82 miles. 6.5 mi A circle has a circumference of 78.578.5 inches. Find the Area of Irregular Figures In the following exercises, find the area of the irregular figure. Round your answers to the nearest hundredth. A geometric shape is shown. It is a horizontal rectangle attached to a vertical rectangle. The top is labeled 6, the height of the horizontal rectangle is labeled 2, the distance from the edge of the horizontal rectangle to the start of the vertical rectangle is 4, the base of the vertical rectangle is 2, the right side of the shape is 4. 16 sq. units A geometric shape is shown. It is an L-shape. The base is labeled 10, the right side 1, the top and left side are each labeled 4. A geometric shape is shown. It is a sideways U-shape. The top is labeled 6, the left side is labeled 6. An inside horizontal piece is labeled 3. Each of the vertical pieces on the right are labeled 2. 30 sq. units A geometric shape is shown. It is a U-shape. The base is labeled 7. The right side is labeled 5. The two horizontal lines at the top and the vertical line on the inside are all labeled 3. A geometric shape is shown. It is a rectangle with a triangle attached to the bottom left side. The top is labeled 4. The right side is labeled 10. The base is labeled 9. The vertical line from the top of the triangle to the top of the rectangle is labeled 3. 57.5 sq. units A trapezoid is shown. The bases are labeled 5 and 10, the height is 5. Two triangles are shown. They appear to be right triangles. The bases are labeled 3, the heights 4, and the longest sides 5. 12 sq. units A geometric shape is shown. It appears to be composed of two triangles. The shared base of both triangles is 8, the heights are both labeled 6. A geometric shape is shown. It is composed of two trapezoids. The base is labeled 10. The height of one trapezoid is 2. The horizontal and vertical sides are all labeled 5. 67.5 sq. units A geometric shape is shown. It is a trapezoid attached to a triangle. The base of the triangle is labeled 6, the height is labeled 5. The height of the trapezoid is 6, one base is 3. A geometric shape is shown. It is a rectangle with a triangle and another rectangle attached. The left side is labeled 8, the bottom is 8, the right side is 13, and the width of the smaller rectangle is 2. 89 sq. units A geometric shape is shown. It is a rectangle with a triangle and another rectangle attached. The left side is labeled 12, the right side 7, the base 6. The width of the smaller rectangle is labeled 1. A geometric shape is shown. It is a rectangle attached to a semi-circle. The base of the rectangle is labeled 5, the height is 7. 44.81 sq. units A geometric shape is shown. It is a rectangle attached to a semi-circle. The base of the rectangle is labeled 10, the height is 6. The portion of the rectangle on the left of the semi-circle is labeled 5, the portion on the right is labeled 2. A geometric shape is shown. A triangle is attached to a semi-circle. The base of the triangle is labeled 4. The height of the triangle and the diameter of the circle are 8. 41.12 sq. units A geometric shape is shown. A triangle is attached to a semi-circle. The height of the triangle is labeled 4. The base of the triangle, also the diameter of the semi-circle, is labeled 4. A geometric shape is shown. It is a rectangle attached to a semi-circle. The base of the rectangle is labeled 5, the height is 7. 35.13 sq. units A geometric shape is shown. A trapezoid is shown with a semi-circle attached to the top. The diameter of the circle, which is also the top of the trapezoid, is labeled 8. The height of the trapezoid is 6. The bottom of the trapezoid is 13. A geometric shape is shown. It is a rectangle with a triangle attached to the top on the left side and a circle attached to the top right corner. The diameter of the circle is labeled 5. The height of the triangle is labeled 5, the base is labeled 4. The height of the rectangle is labeled 6, the base 11. 95.625 sq. units A geometric shape is shown. It is a trapezoid with a triangle attached to the top, and a circle attached to the triangle. The diameter of the circle is 4. The height of the triangle is 5, the base of the triangle, which is also the top of the trapezoid, is 6. The bottom of the trapezoid is 9. The height of the trapezoid is 7. In the following exercises, solve. A city park covers one block plus parts of four more blocks, as shown. The block is a square with sides 250250 feet long, and the triangles are isosceles right triangles. Find the area of the park. A square is shown with four triangles coming off each side. 187,500 sq. ft A gift box will be made from a rectangular piece of cardboard measuring 1212 inches by 2020 inches, with squares cut out of the corners of the sides, as shown. The sides of the squares are 33 inches. Find the area of the cardboard after the corners are cut out. A rectangle is shown. Each corner has a gray shaded square. There are dotted lines drawn across the side of each square attached to the next square. Perry needs to put in a new lawn. His lot is a rectangle with a length of 120120 feet and a width of 100100 feet. The house is rectangular and measures 5050 feet by 4040 feet. His driveway is rectangular and measures 2020 feet by 3030 feet, as shown. Find the area of Perry’s lawn. A rectangular lot is shown. In it is a home shaped like a rectangle attached to a rectangular driveway. 9400 sq. ft Denise is planning to put a deck in her back yard. The deck will be a 20-ft\text{20-ft} by 12-ft\text{12-ft} rectangle with a semicircle of diameter 66 feet, as shown below. Find the area of the deck. A picture of a deck is shown. It is shaped like a rectangle with a semi-circle attached to the top on the left side.

Everyday Math

Area of a Tabletop Yuki bought a drop-leaf kitchen table. The rectangular part of the table is a 1-ft\text{1-ft} by 3-ft\text{3-ft} rectangle with a semicircle at each end, as shown. ⓐ Find the area of the table with one leaf up. ⓑ Find the area of the table with both leaves up. An image of a table is shown. There is a rectangular portion attached to a semi-circular portion. There is another semi-circular leaf folded down on the other side of the rectangle. ⓐ 6.5325 sq. ft ⓑ 10.065 sq. ft Painting Leora wants to paint the nursery in her house. The nursery is an 8-ft\text{8-ft} by 10-ft\text{10-ft} rectangle, and the ceiling is 88 feet tall. There is a 3-ft\text{3-ft} by 6.5-ft\text{6.5-ft} door on one wall, a 3-ft\text{3-ft} by 6.5-ft\text{6.5-ft} closet door on another wall, and one 4-ft\text{4-ft} by 3.5-ft\text{3.5-ft} window on the third wall. The fourth wall has no doors or windows. If she will only paint the four walls, and not the ceiling or doors, how many square feet will she need to paint?

Writing Exercises

Describe two different ways to find the area of this figure, and then show your work to make sure both ways give the same area. A geometric shape is shown. It is a vertical rectangle attached to a horizontal rectangle. The width of the vertical rectangle is 3, the left side is labeled 6, the bottom is labeled 9, and the width of the horizontal rectangle is labeled 3. The top of the horizontal rectangle is labeled 6, and the distance from the top of that rectangle to the top of the other rectangle is labeled 3. Answers will vary. A circle has a diameter of 1414 feet. Find the area of the circle ⓐ using 3.143.14 for π\pi ⓑ using 227\frac{22}{7} for \pi\text{\pi }. ⓒ Which calculation to do prefer? Why?

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. . ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?

Practice Makes Perfect

Find Volume and Surface Area of Rectangular Solids In the following exercises, find ⓐ the volume and ⓑ the surface area of the rectangular solid with the given dimensions. length 22 meters, width 1.51.5 meters, height 33 meters ⓐ 9 cu. m ⓑ 27 sq. m length 55 feet, width 88 feet, height 2.52.5 feet length 3.53.5 yards, width 2.12.1 yards, height 2.42.4 yards ⓐ 17.64 cu. yd. ⓑ 41.58 sq. yd. length 8.88.8 centimeters, width 6.56.5 centimeters, height 4.24.2 centimeters In the following exercises, solve. Moving van A rectangular moving van has length 1616 feet, width 88 feet, and height 88 feet. Find its ⓐ volume and ⓑ surface area. ⓐ 1,024 cu. ft ⓑ 640 sq. ft Gift box A rectangular gift box has length 2626 inches, width 1616 inches, and height 44 inches. Find its ⓐ volume and ⓑ surface area. Carton A rectangular carton has length 21.321.3 cm, width 24.224.2 cm, and height 6.56.5 cm. Find its ⓐ volume and ⓑ surface area. ⓐ 3,350.49 cu. cm ⓑ 1,622.42 sq. cm Shipping container A rectangular shipping container has length 22.822.8 feet, width 8.58.5 feet, and height 8.28.2 feet. Find its ⓐ volume and ⓑ surface area. In the following exercises, find ⓐ the volume and ⓑ the surface area of the cube with the given side length. 55 centimeters ⓐ 125 cu. cm ⓑ 150 sq. cm 66 inches 10.410.4 feet ⓐ 1124.864 cu. ft. ⓑ 648.96 sq. ft 12.512.5 meters In the following exercises, solve. Science center Each side of the cube at the Discovery Science Center in Santa Ana is 6464 feet long. Find its ⓐ volume and ⓑ surface area. ⓐ 262,144 cu. ft ⓑ 24,576 sq. ft Museum A cube-shaped museum has sides 4545 meters long. Find its ⓐ volume and ⓑ surface area. Base of statue The base of a statue is a cube with sides 2.82.8 meters long. Find its ⓐ volume and ⓑ surface area. ⓐ 21.952 cu. m ⓑ 47.04 sq. m Tissue box A box of tissues is a cube with sides 4.5 inches long. Find its ⓐ volume and ⓑ surface area. Find the Volume and Surface Area of Spheres In the following exercises, find ⓐ the volume and ⓑ the surface area of the sphere with the given radius. Round answers to the nearest hundredth. 33 centimeters ⓐ 113.04 cu. cm ⓑ 113.04 sq. cm 99 inches 7.57.5 feet ⓐ 1,766.25 cu. ft ⓑ 706.5 sq. ft 2.12.1 yards In the following exercises, solve. Round answers to the nearest hundredth. Exercise ball An exercise ball has a radius of 1515 inches. Find its ⓐ volume and ⓑ surface area. ⓐ 14,130 cu. in. ⓑ 2,826 sq. in. Balloon ride The Great Park Balloon is a big orange sphere with a radius of 3636 feet . Find its ⓐ volume and ⓑ surface area. Golf ball A golf ball has a radius of 4.54.5 centimeters. Find its ⓐ volume and ⓑ surface area. ⓐ 381.51 cu. cm ⓑ 254.34 sq. cm Baseball A baseball has a radius of 2.92.9 inches. Find its ⓐ volume and ⓑ surface area. Find the Volume and Surface Area of a Cylinder In the following exercises, find ⓐ the volume and ⓑ the surface area of the cylinder with the given radius and height. Round answers to the nearest hundredth. radius 33 feet, height 99 feet ⓐ 254.34 cu. ft ⓑ 226.08 sq. ft radius 55 centimeters, height 1515 centimeters radius 1.51.5 meters, height 4.24.2 meters ⓐ 29.673 cu. m ⓑ 53.694 sq. m radius 1.31.3 yards, height 2.82.8 yards In the following exercises, solve. Round answers to the nearest hundredth. Coffee can A can of coffee has a radius of 55 cm and a height of 1313 cm. Find its ⓐ volume and ⓑ surface area. ⓐ 1,020.5 cu. cm ⓑ 565.2 sq. cm Snack pack A snack pack of cookies is shaped like a cylinder with radius 44 cm and height 33 cm. Find its ⓐ volume and ⓑ surface area. Barber shop pole A cylindrical barber shop pole has a diameter of 66 inches and height of 2424 inches. Find its ⓐ volume and ⓑ surface area. ⓐ 678.24 cu. in. ⓑ 508.68 sq. in. Architecture A cylindrical column has a diameter of 88 feet and a height of 2828 feet. Find its ⓐ volume and ⓑ surface area. Find the Volume of Cones In the following exercises, find the volume of the cone with the given dimensions. Round answers to the nearest hundredth. height 99 feet and radius 22 feet 37.68 cu. ft height 88 inches and radius 66 inches height 12.412.4 centimeters and radius 55 cm 324.47 cu. cm height 15.215.2 meters and radius 44 meters In the following exercises, solve. Round answers to the nearest hundredth. Teepee What is the volume of a cone-shaped teepee tent that is 1010 feet tall and 1010 feet across at the base? 261.67 cu. ft Popcorn cup What is the volume of a cone-shaped popcorn cup that is 88 inches tall and 66 inches across at the base? Silo What is the volume of a cone-shaped silo that is 5050 feet tall and 7070 feet across at the base? 64,108.33 cu. ft Sand pile What is the volume of a cone-shaped pile of sand that is 1212 meters tall and 3030 meters across at the base?

Everyday Math

Street light post The post of a street light is shaped like a truncated cone, as shown in the picture below. It is a large cone minus a smaller top cone. The large cone is 3030 feet tall with base radius 11 foot. The smaller cone is 1010 feet tall with base radius of 0.50.5 feet. To the nearest tenth, ⓐ find the volume of the large cone. ⓑ find the volume of the small cone. ⓒ find the volume of the post by subtracting the volume of the small cone from the volume of the large cone. An image of a cone is shown. There is a dark dotted line at the top indicating a smaller cone. ⓐ 31.4 cu. ft ⓑ 2.6 cu. ft ⓒ 28.8 cu. ft Ice cream cones A regular ice cream cone is 4 inches tall and has a diameter of 2.52.5 inches. A waffle cone is 77 inches tall and has a diameter of 3.253.25 inches. To the nearest hundredth, ⓐ find the volume of the regular ice cream cone. ⓑ find the volume of the waffle cone. ⓒ how much more ice cream fits in the waffle cone compared to the regular cone?

Writing Exercises

The formulas for the volume of a cylinder and a cone are similar. Explain how you can remember which formula goes with which shape. Answers will vary. Which has a larger volume, a cube of sides of 88 feet or a sphere with a diameter of 88 feet? Explain your reasoning.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. . ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

Practice Makes Perfect

Make Unit Conversions in the U.S. System In the following exercises, convert the units. A park bench is 66 feet long. Convert the length to inches. A floor tile is 22 feet wide. Convert the width to inches. 24 inches A ribbon is 1818 inches long. Convert the length to feet. Carson is 4545 inches tall. Convert his height to feet. 3.75 feet Jon is 66 feet 44 inches tall. Convert his height to inches. Faye is 44 feet 1010 inches tall. Convert her height to inches. 58 inches A football field is 160160 feet wide. Convert the width to yards. On a baseball diamond, the distance from home plate to first base is 3030 yards. Convert the distance to feet. 90 feet Ulises lives 1.51.5 miles from school. Convert the distance to feet. Denver, Colorado, is 5,1835,183 feet above sea level. Convert the height to miles. 0.98 miles A killer whale weighs 4.64.6 tons. Convert the weight to pounds. Blue whales can weigh as much as 150150 tons. Convert the weight to pounds. 300,000 pounds An empty bus weighs 35,00035,000 pounds. Convert the weight to tons. At take-off, an airplane weighs 220,000220,000 pounds. Convert the weight to tons. 110 tons The voyage of the Mayflower took 22 months and 55 days. Convert the time to days. Lynn’s cruise lasted 66 days and 1818 hours. Convert the time to hours. 162 hours Rocco waited 1121\frac{1}{2} hours for his appointment. Convert the time to seconds. Misty’s surgery lasted 2142\frac{1}{4} hours. Convert the time to seconds. 8100 seconds How many teaspoons are in a pint? How many tablespoons are in a gallon? 256 tablespoons JJ’s cat, Posy, weighs 1414 pounds. Convert her weight to ounces. April’s dog, Beans, weighs 88 pounds. Convert his weight to ounces. 128 ounces Baby Preston weighed 77 pounds 33 ounces at birth. Convert his weight to ounces. Baby Audrey weighed 66 pounds 1515 ounces at birth. Convert her weight to ounces. 111 ounces Crista will serve 2020 cups of juice at her son’s party. Convert the volume to gallons. Lance needs 500500 cups of water for the runners in a race. Convert the volume to gallons. 31.25 gallons Use Mixed Units of Measurement in the U.S. System In the following exercises, solve and write your answer in mixed units. Eli caught three fish. The weights of the fish were 22 pounds 44 ounces, 11 pound 1111 ounces, and 44 pounds 1414 ounces. What was the total weight of the three fish? Judy bought 11 pound 66 ounces of almonds, 22 pounds 33 ounces of walnuts, and 88 ounces of cashews. What was the total weight of the nuts? 4 lbs. 1 oz. One day Anya kept track of the number of minutes she spent driving. She recorded trips of 45,10,8,65,20,and 35 minutes.45,10,8,65,20,\text{and 35 minutes.} How much time (in hours and minutes) did Anya spend driving? Last year Eric went on 66 business trips. The number of days of each was 5,2,8,12,6,and 3.5,2,8,12,6,\text{and 3.} How much time (in weeks and days) did Eric spend on business trips last year? 5 weeks and 1 day Renee attached a 6-foot - 6-inch\text{6-foot - 6-inch} extension cord to her computer’s 3-foot - 8-inch\text{3-foot - 8-inch} power cord. What was the total length of the cords? Fawzi’s SUV is 66 feet 44 inches tall. If he puts a 2-foot - 10-inch\text{2-foot - 10-inch} box on top of his SUV, what is the total height of the SUV and the box? 9 ft 2 in Leilani wants to make 88 placemats. For each placemat she needs 1818 inches of fabric. How many yards of fabric will she need for the 88 placemats? Mireille needs to cut 2424 inches of ribbon for each of the 1212 girls in her dance class. How many yards of ribbon will she need altogether? 8 yards Make Unit Conversions in the Metric System In the following exercises, convert the units. Ghalib ran 55 kilometers. Convert the length to meters. Kitaka hiked 88 kilometers. Convert the length to meters. 8000 meters Estrella is 1.551.55 meters tall. Convert her height to centimeters. The width of the wading pool is 2.452.45 meters. Convert the width to centimeters. 245 centimeters Mount Whitney is 3,0723,072 meters tall. Convert the height to kilometers. The depth of the Mariana Trench is 10,91110,911 meters. Convert the depth to kilometers. 10.911 kilometers June’s multivitamin contains 1,5001,500 milligrams of calcium. Convert this to grams. A typical ruby-throated hummingbird weights 33 grams. Convert this to milligrams. 3000 milligrams One stick of butter contains 91.691.6 grams of fat. Convert this to milligrams. One serving of gourmet ice cream has 2525 grams of fat. Convert this to milligrams. 25,000 milligrams The maximum mass of an airmail letter is 22 kilograms. Convert this to grams. Dimitri’s daughter weighed 3.83.8 kilograms at birth. Convert this to grams. 3800 grams A bottle of wine contained 750750 milliliters. Convert this to liters. A bottle of medicine contained 300300 milliliters. Convert this to liters. 0.3 liters Use Mixed Units of Measurement in the Metric System In the following exercises, solve and write your answer in mixed units. Matthias is 1.81.8 meters tall. His son is 8989 centimeters tall. How much taller, in centimeters, is Matthias than his son? Stavros is 1.61.6 meters tall. His sister is 9595 centimeters tall. How much taller, in centimeters, is Stavros than his sister? 65 centimeters A typical dove weighs 345345 grams. A typical duck weighs 1.21.2 kilograms. What is the difference, in grams, of the weights of a duck and a dove? Concetta had a 2-kilogram\text{2-kilogram} bag of flour. She used 180180 grams of flour to make biscotti. How many kilograms of flour are left in the bag? 1.82 kilograms Harry mailed 55 packages that weighed 420420 grams each. What was the total weight of the packages in kilograms? One glass of orange juice provides 560560 milligrams of potassium. Linda drinks one glass of orange juice every morning. How many grams of potassium does Linda get from her orange juice in 3030 days? 16.8 grams Jonas drinks 200200 milliliters of water 88 times a day. How many liters of water does Jonas drink in a day? One serving of whole grain sandwich bread provides 66 grams of protein. How many milligrams of protein are provided by 77 servings of whole grain sandwich bread? 42,000 milligrams Convert Between U.S. and Metric Systems In the following exercises, make the unit conversions. Round to the nearest tenth. Bill is 7575 inches tall. Convert his height to centimeters. Frankie is 4242 inches tall. Convert his height to centimeters. 106.7 centimeters Marcus passed a football 2424 yards. Convert the pass length to meters. Connie bought 99 yards of fabric to make drapes. Convert the fabric length to meters. 8.2 meters Each American throws out an average of 1,6501,650 pounds of garbage per year. Convert this weight to kilograms. An average American will throw away 90,00090,000 pounds of trash over his or her lifetime. Convert this weight to kilograms. 41,500 kilograms A 5K\text{5K} run is 55 kilometers long. Convert this length to miles. Kathryn is 1.61.6 meters tall. Convert her height to feet. 5.2 feet Dawn’s suitcase weighed 2020 kilograms. Convert the weight to pounds. Jackson’s backpack weighs 1515 kilograms. Convert the weight to pounds. 33 pounds Ozzie put 1414 gallons of gas in his truck. Convert the volume to liters. Bernard bought 88 gallons of paint. Convert the volume to liters. 30.4 liters Convert between Fahrenheit and Celsius In the following exercises, convert the Fahrenheit temperature to degrees Celsius. Round to the nearest tenth. 86\text{^\circ F} 77\text{^\circ F} 25°C 104\text{^\circ F} 14\text{^\circ F} −10°C 72\text{^\circ F} 4\text{^\circ F} −15.5°C 0\text{^\circ F} 120\text{^\circ F} 48.9°C In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth. 5\text{^\circ C} 25\text{^\circ C} 77°F -10\text{^\circ C} -15\text{^\circ C} 5°F 22\text{^\circ C} 8\text{^\circ C} 46.4°F 43\text{^\circ C} 16\text{^\circ C} 60.8°F

Everyday Math

Nutrition Julian drinks one can of soda every day. Each can of soda contains 4040 grams of sugar. How many kilograms of sugar does Julian get from soda in 11 year? Reflectors The reflectors in each lane-marking stripe on a highway are spaced 1616 yards apart. How many reflectors are needed for a one-mile-long stretch of highway? 110 reflectors

Writing Exercises

Some people think that 65\text{^\circ } to 75\text{^\circ } Fahrenheit is the ideal temperature range. ⓐ What is your ideal temperature range? Why do you think so? ⓑ Convert your ideal temperatures from Fahrenheit to Celsius. ⓐ Did you grow up using the U.S. customary or the metric system of measurement? ⓑ Describe two examples in your life when you had to convert between systems of measurement. ⓒ Which system do you think is easier to use? Explain.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. . ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next chapter? Why or why not?

Chapter Review Exercises

Rational and Irrational Numbers

In the following exercises, write as the ratio of two integers. 66 −5-5 −51\frac{-5}{1} 2.92.9 1.81.8 1810\frac{18}{10} In the following exercises, determine which of the numbers is rational. 0.42,0.3-,2.56813…0.42,0.\stackrel{\text{-}}{\text{3}},2.56813\dots 0.75319…,0.16-,1.95\text{0.75319\ldots },0.\stackrel{\text{-}}{16},1.95 0.16-,1.950.\stackrel{\text{-}}{16},1.95 In the following exercises, identify whether each given number is rational or irrational. ⓐ 49\sqrt{49} ⓑ 55\sqrt{55} ⓐ 72\sqrt{72} ⓑ 64\sqrt{64} ⓐ irrational ⓑ rational In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers. −9,0,0.361....,89,16,9-9,0,\text{0.361....},\frac{8}{9},\sqrt{16},9 −5,−214,−4,0.25-,135,4-5,-2\frac{1}{4},-\sqrt{4},0.\stackrel{\text{-}}{25},\frac{13}{5},4 ⓐ 44 ⓑ −5,−4,4-5,-\sqrt{4},4 ⓒ −5,−214,−4,0.25-,135,4-5,-2\frac{1}{4},-\sqrt{4},0.\stackrel{\text{-}}{25},\frac{13}{5},4 ⓓ none ⓔ −5,−214,−4,0.25-,135,4-5,-2\frac{1}{4},-\sqrt{4},0.\stackrel{\text{-}}{25},\frac{13}{5},4

Commutative and Associative Properties

In the following exercises, use the commutative property to rewrite the given expression. 6+4=____ -14\cdot 5=____ −14·5 = 5(−14) 3n=____ a+8=____ a + 8 = 8 + a In the following exercises, use the associative property to rewrite the given expression. \left(13\cdot 5\right)\cdot 2=_____ \left(22+7\right)+3=_____ (22 + 7) + 3 = 22 + (7 + 3) \left(4+9x\right)+x=_____ \frac{1}{2}\left(22y\right)=_____ 12(22y)=(12⋅22)y\frac{1}{2}\left(22y\right)=\left(\frac{1}{2}\cdot 22\right)y In the following exercises, evaluate each expression for the given value. If y=1112y=\frac{11}{12}, evaluate: ⓐ y+0.7+(−y)y+0.7+\left(-y\right) ⓑ y+(−y)+0.7y+\left(-y\right)+0.7 If z=−53z=-\frac{5}{3}, evaluate: ⓐ z+5.39+(−z)z+5.39+\left(-z\right) ⓑ z+(−z)+5.39z+\left(-z\right)+5.39 ⓐ 5.39 ⓑ 5.39 If k=65k=65, evaluate: ⓐ 49(94k)\frac{4}{9}\left(\frac{9}{4}k\right) ⓑ (49⋅94)k\left(\frac{4}{9}\cdot \frac{9}{4}\right)k If m=−13m=-13, evaluate: ⓐ −25(52m)-\frac{2}{5}\left(\frac{5}{2}m\right) ⓑ (−25⋅52)m\left(-\frac{2}{5}\cdot \frac{5}{2}\right)m ⓐ 13 ⓑ 13 In the following exercises, simplify using the commutative and associative properties. 6y+37+(−6y)6y+37+\left(-6y\right) 14+1115+(−14)\frac{1}{4}+\frac{11}{15}+\left(-\frac{1}{4}\right) 1115\frac{11}{15} 1411⋅359⋅1411\frac{14}{11}\cdot \frac{35}{9}\cdot \frac{14}{11} −18⋅15⋅29-18\cdot 15\cdot \frac{2}{9} −60 (712+45)+15\left(\frac{7}{12}+\frac{4}{5}\right)+\frac{1}{5} (3.98d+0.75d)+1.25d\left(3.98d+0.75d\right)+1.25d 5.98 d −12(4m)-12\left(4m\right) 30(56q)30\left(\frac{5}{6}q\right) 25 q 11x+8y+16x+15y11x+8y+16x+15y 52m+(−20n)+(−18m)+(−5n)52m+\left(-20n\right)+\left(-18m\right)+\left(-5n\right) 34 m + (−25 n)

Distributive Property

In the following exercises, simplify using the distributive property. 7(x+9)7\left(x+9\right) 9(u−4)9\left(u - 4\right) 9y − 36 −3(6m−1)-3\left(6m - 1\right) −8(−7a−12)-8\left(-7a - 12\right) 56a + 96 13(15n−6)\frac{1}{3}\left(15n - 6\right) (y+10)⋅p\left(y+10\right)\cdot p yp + 10p (a−4)−(6a+9)\left(a - 4\right)-\left(6a+9\right) 4(x+3)−8(x−7)4\left(x+3\right)-8\left(x - 7\right) −4x + 68 In the following exercises, evaluate using the distributive property. If u=2u=2, evaluate ⓐ 3(8u+9)and3\left(8u+9\right)\text{and} ⓑ 3⋅8u+3⋅93\cdot 8u+3\cdot 9 to show that 3(8u+9)=3⋅8u+3⋅93\left(8u+9\right)=3\cdot 8u+3\cdot 9 If n=78n=\frac{7}{8}, evaluate ⓐ 8(n+14)8\left(n+\frac{1}{4}\right) and ⓑ 8⋅n+8⋅148\cdot n+8\cdot \frac{1}{4} to show that 8(n+14)=8⋅n+8⋅148\left(n+\frac{1}{4}\right)=8\cdot n+8\cdot \frac{1}{4} ⓐ 9 ⓑ 9 If d=14d=14, evaluate ⓐ −100(0.1d+0.35)-100\left(0.1d+0.35\right) and ⓑ −100⋅(0.1d)+(−100)(0.35)-100\cdot \left(0.1d\right)+\left(-100\right)\left(0.35\right) to show that −100(0.1d+0.35)=−100⋅(0.1d)+(−100)(0.35)-100\left(0.1d+0.35\right)=-100\cdot \left(0.1d\right)+\left(-100\right)\left(0.35\right) If y=−18y=-18, evaluate ⓐ −(y−18)-\left(y - 18\right) and ⓑ −y+18-y+18 to show that −(y−18)=−y+18-\left(y - 18\right)=-y+18 ⓐ 36 ⓑ 36

Properties of Identities, Inverses, and Zero

In the following exercises, identify whether each example is using the identity property of addition or multiplication. −35(1)=−35-35\left(1\right)=-35 29+0=2929+0=29 identity property of addition (6x+0)+4x=6x+4x\left(6x+0\right)+4x=6x+4x 9⋅1+(−3)=9+(−3)9\cdot 1+\left(-3\right)=9+\left(-3\right) identity property of multiplication In the following exercises, find the additive inverse. −32-32 19.419.4 −19.4 35\frac{3}{5} −715-\frac{7}{15} 715\frac{7}{15} In the following exercises, find the multiplicative inverse. 92\frac{9}{2} −5-5 −15-\frac{1}{5} 110\frac{1}{10} −49-\frac{4}{9} −94-\frac{9}{4} In the following exercises, simplify. 83⋅083\cdot 0 09\frac{0}{9} 0 50\frac{5}{0} 0÷230\div \frac{2}{3} 0 43+39+(−43)43+39+\left(-43\right) (n+6.75)+0.25\left(n+6.75\right)+0.25 n + 7 513⋅57⋅135\frac{5}{13}\cdot 57\cdot \frac{13}{5} 16⋅17⋅12\frac{1}{6}\cdot 17\cdot 12 34 23⋅28⋅37\frac{2}{3}\cdot 28\cdot \frac{3}{7} 9(6x−11)+159\left(6x - 11\right)+15 54x − 84

Systems of Measurement

In the following exercises, convert between U.S. units. Round to the nearest tenth. A floral arbor is 77 feet tall. Convert the height to inches. A picture frame is 4242 inches wide. Convert the width to feet. 3.5 feet Kelly is 55 feet 44 inches tall. Convert her height to inches. A playground is 4545 feet wide. Convert the width to yards. 15 yards The height of Mount Shasta is 14,17914,179 feet. Convert the height to miles. Shamu weighs 4.54.5 tons. Convert the weight to pounds. 9000 pounds The play lasted 1341\frac{3}{4} hours. Convert the time to minutes. How many tablespoons are in a quart? 64 tablespoons Naomi’s baby weighed 55 pounds 1414 ounces at birth. Convert the weight to ounces. Trinh needs 3030 cups of paint for her class art project. Convert the volume to gallons. 1.9 gallons In the following exercises, solve, and state your answer in mixed units. John caught 44 lobsters. The weights of the lobsters were 11 pound 99 ounces, 11 pound 1212 ounces, 44 pounds 22 ounces, and 22 pounds 1515 ounces. What was the total weight of the lobsters? Every day last week, Pedro recorded the amount of time he spent reading. He read for 50,25,83,45,32,60,and13550,25,83,45,32,60,\text{and}135 minutes. How much time, in hours and minutes, did Pedro spend reading? 7 hours 10 minutes Fouad is 66 feet 22 inches tall. If he stands on a rung of a ladder 88 feet 1010 inches high, how high off the ground is the top of Fouad’s head? Dalila wants to make pillow covers. Each cover takes 3030 inches of fabric. How many yards and inches of fabric does she need for 44 pillow covers? 3 yards, 12 inches In the following exercises, convert between metric units. Donna is 1.71.7 meters tall. Convert her height to centimeters. Mount Everest is 8,8508,850 meters tall. Convert the height to kilometers. 8.85 kilometers One cup of yogurt contains 488488 milligrams of calcium. Convert this to grams. One cup of yogurt contains 1313 grams of protein. Convert this to milligrams. 13,000 milligrams Sergio weighed 2.92.9 kilograms at birth. Convert this to grams. A bottle of water contained 650650 milliliters. Convert this to liters. 0.65 liters In the following exercises, solve. Minh is 22 meters tall. His daughter is 8888 centimeters tall. How much taller, in meters, is Minh than his daughter? Selma had a 1-liter\text{1-liter} bottle of water. If she drank 145145 milliliters, how much water, in milliliters, was left in the bottle? 855 milliliters One serving of cranberry juice contains 3030 grams of sugar. How many kilograms of sugar are in 3030 servings of cranberry juice? One ounce of tofu provides 22 grams of protein. How many milligrams of protein are provided by 55 ounces of tofu? 10,000 milligrams In the following exercises, convert between U.S. and metric units. Round to the nearest tenth. Majid is 6969 inches tall. Convert his height to centimeters. A college basketball court is 8484 feet long. Convert this length to meters. 25.6 meters Caroline walked 2.52.5 kilometers. Convert this length to miles. Lucas weighs 7878 kilograms. Convert his weight to pounds. 171.6 pounds Steve’s car holds 5555 liters of gas. Convert this to gallons. A box of books weighs 2525 pounds. Convert this weight to kilograms. 11.4 kilograms In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth. 95\text{^\circ F} 23\text{^\circ F} −5°C 20\text{^\circ F} 64\text{^\circ F} 17.8°C In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth. 30\text{^\circ C} -5\text{^\circ C} 23°F -12\text{^\circ C} 24\text{^\circ C} 75.2°F

Chapter Practice Test

For the numbers 0.18349…,0.2-,1.67\text{0.18349\ldots },0.\stackrel{\text{-}}{\text{2}},1.67, list the ⓐ rational numbers and ⓑ irrational numbers. Is 144\sqrt{144} rational or irrational? 144=12therefore rational.\sqrt{144}=12\text{therefore rational.} From the numbers −4,−112,0,58,2,7-4,-1\frac{1}{2},0,\frac{5}{8},\sqrt{2},7, which are ⓐ integers ⓑ rational ⓒ irrational ⓓ real numbers? Rewrite using the commutative property: x\cdot 14=_________ x·14 = 14·x Rewrite the expression using the associative property: \left(y+6\right)+3=_______________ Rewrite the expression using the associative property: \left(8\cdot 2\right)\cdot 5=___________ (8·2)·3 = 8·(2·3) Evaluate 316(163n)\frac{3}{16}\left(\frac{16}{3}n\right) when n=42n=42. For the number 25\frac{2}{5} find the ⓐ additive inverse ⓑ multiplicative inverse. ⓐ −25-\frac{2}{5} ⓑ 52\frac{5}{2} In the following exercises, simplify the given expression. 34(−29)(43)\frac{3}{4}\left(-29\right)\left(\frac{4}{3}\right) −3+15y+3-3+15y+3 15y (1.27q+0.25q)+0.75q\left(1.27q+0.25q\right)+0.75q (815+29)+79\left(\frac{8}{15}+\frac{2}{9}\right)+\frac{7}{9} 2315\frac{23}{15} −18(32n)-18\left(\frac{3}{2}n\right) 14y+(−6z)+16y+2z14y+\left(-6z\right)+16y+2z 30y − 4z 9(q+9)9\left(q+9\right) 6(5x−4)6\left(5x - 4\right) 30x − 24 −10(0.4n+0.7)-10\left(0.4n+0.7\right) 14(8a+12)\frac{1}{4}\left(8a+12\right) 2a + 3 m(n+2)m\left(n+2\right) 8(6p−1)+2(9p+3)8\left(6p - 1\right)+2\left(9p+3\right) 66p − 2 (12a+4)−(9a+6)\left(12a+4\right)-\left(9a+6\right) 08\frac{0}{8} 0 4.50\frac{4.5}{0} 0÷(23)0\div \left(\frac{2}{3}\right) 0 In the following exercises, solve using the appropriate unit conversions. Azize walked 4124\frac{1}{2} miles. Convert this distance to feet. (1 mile=5,280 feet).\text{(1 mile}=\text{5,280 feet).} One cup of milk contains 276276 milligrams of calcium. Convert this to grams. (1 milligram=0.001 gram)\text{(1 milligram}=\text{0.001 gram)} .276 grams Larry had 55 phone customer phone calls yesterday. The calls lasted 28,44,9,75,and5528,44,9,75,\text{and}55 minutes. How much time, in hours and minutes, did Larry spend on the phone? (1 hour=60 minutes)\text{(1 hour}=\text{60 minutes)} Janice ran 1515 kilometers. Convert this distance to miles. Round to the nearest hundredth of a mile. (1 mile=1.61 kilometers)\text{(1 mile}=\text{1.61 kilometers)} 9.317 miles Yolie is 6363 inches tall. Convert her height to centimeters. Round to the nearest centimeter. (1 inch=2.54 centimeters)\text{(1 inch}=\text{2.54 centimeters)} Use the formula F=95C+32F=\frac{9}{5}C+32 to convert 35\text{^\circ C} to degrees F\text{F} 95°F  

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