1ACE Exercise 7 Investigation 1 7. This closed rectangular box does not have square ends. 2 cm 1 cm 4 cm a. What are the dimensions of the box? Height: Length: Width: b. Sketch 2 nets for the box. Refer back to Problem 1.2 for an example. c. Find the area, in square centimeters, of each net. Net 1 Area: Net 2 Area: d. Find the total area of all the faces of the box. a. Area of all faces of the box: a. How does your answer compare to the areas you found in part (c)? 107
2ACE Exercise 5 Investigation 2 5. Each of these boxes holds 36 ping-pong balls. 48 cm 16 cm 8 cm 24 cm 24 cm 8 cm 4 cm a. Without figuring, which box has the least surface area? HINT Remember that surface area of a box is the total area of all its faces. a. Why does it have the least surface area? b. Check your guess by finding the surface area of each box. a. Box A: a. Box B: a. Box C: 108
3ACE Exercise 22 Investigation 3 22. A popcorn vendor needs to order popcorn boxes. The vendor must decide between a cylindrical box and a rectangular box. The cylindrical box has a height of 20 centimeters and a radius of 7 centimeters. 7 cm 20 cm The rectangular box has a height of 20 centimeters and a square base with 12-centimeter sides. 20 cm The price of each box is based on the amount of material needed to make the box. The vendor plans to charge $2.75 for popcorn, regardless of the shape of the box. a. Make a sketch of each box. Label the dimensions. b. cylindrical box rectangular box 109
3ACE Exercise 22 (continued) Investigation 3 b. Find the volume and surface area of each box. a. Volume of cylindrical box: a. Surface area of cylindrical box: a. Volume of rectangular box: a. Surface area of rectangular box: c. Which box would you choose? a. Give the reasons for your choice. a. Reason 1: a. Reason 2: a. Other reasons? a. What additional information might help you make a better decision? 110
4ACE Exercise 11 Investigation 4 11. Find the volume of each shape. a. b. c. 3 cm 3 cm 6 cm 6 cm 3 cm Cylinder Volume = Cone Volume = Sphere Volume = d. How do the volumes of the three shapes compare? a. Do any of the containers have the same volume? a. Which has the largest volume? a. Which has the smallest volume? a. How many times bigger is the largest volume compared to the smallest volume? 111
5ACE Exercise 17 Investigation 5 17. For every ton of paper that is recycled, about 17 trees and 3.3 yd 3 of landfill space are saved. In the United States, the equivalent of 500,000 trees are used each week to produce the Sunday papers. Suppose all the Sunday papers this week are made from recycled paper. How much landfill is saved? HINT When 1 ton of paper is recycled, it saves 17 trees and 3.3 yd 3 of landfill. So, if you had 2 tons of recycled paper, you would save 34 trees (2 x 17) and 6.6 yd 3 (2 x 3.3) of landfill (see table). Recycled Paper Tons of Paper Trees Saved Landfill Saved (yd 3 ) 1 17 3.3 2 3 34 61 6.6 9.9 500,000 If you continued the chart and had 3 tons of recycled paper, you would save 61 trees (3 x 17) and 9.9 yd 3 (3 x 3.3) of landfill. If the chart continued until you had 500,000 trees, how much landfill would you save? 112
Partner Quiz Goop and Gunk, Inc. sells oil, grease, and other compounds. Cleaning compounds can be purchased in 2 sizes of cylindrical containers, a small size for the home and a large size for businesses. The home size can has a radius of 2.75 inches and a height of 8.25 inches. The business size can has a radius of 11 inches and a height of 33 inches. 1. How many square inches of aluminum are needed to make the home size can? (Assume the cans have flat bottoms and tops.) Goop & 8.25 in. Gunk Business Size r = 11 inches h = 33 inches 2.75 in. Goop & Gunk Home Size r = 2.75 inches h = 8.25 inches Find the surface area of the home size can (sides, top, and bottom). HINT Remember to label your answers with the appropriate label square inches, cubic inches, etc. Using the surface area, how many square inches of aluminum are needed to make the home size can? 2. How many square inches of aluminum are needed to make the business size can? Find the surface area of the business size can (sides, top, and bottom). Using the surface area, how many square inches of aluminum are needed to make the business size can? 3. How many cubic inches of cleaning compound will the home size can hold? To answer this, you need to find the volume of the home size can. 113
Partner Quiz (continued) 4. How many cubic inches of cleaning compound will the business size can hold? To answer this, you need to find the volume of the business size can? 5. If a home size can sells for 85, what should the price of the business size can be if the company wants to base the price on the amount of cleaning compound the can will hold? What is the amount of compound the home size can will hold? What is the amount of compound the business size can will hold? How many times more compound fits in the business size compared to the home size can? What should the price of the business size can be? 6. Goop and Gunk, Inc. is also planning to package their home size cleaning compound in a triangular prism. They want the triangular prism to hold the same amount of cleaning compound as the home size can. Give the dimensions of a triangular prism that will hold the same amount. Look back at your other answers to find the amount of compound the home size can will hold. How do you find how much a triangular prism can hold? HINT How do you find the volume of a triangular prism? 114