Section C Practice Problems

Problem 1

What is the volume of this figure? Explain or show your reasoning.

Rectangular prism with additional cubes attached to front.
Prism: 3 cubes by 5 cubes by 4 cubes. Additional cubes: two of three cubes attached to front, one cube on top of row of three cubes. 

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Solution
64 cubic units. Sample response: There are 4 layers of 3 by 5 so that is 4×(3×5)4 \times (3 \times 5) or 60, and there are 4 extra cubes in front.

Problem 2

Find the volume of the figure. Explain or show your reasoning.

6-sided rectangular prism.
6-sided rectangular prism. Straight sides. All side lengths meet at right angles. Side lengths. Bottom, unlabeled length. Right side rises unlabeled length, then goes left 3 inches, goes up 2 inches, then goes left an 5 inches, then goes down 3 inches. Width shown as 4 inches.

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Solution
72 cubic inches. Sample response: I cut the figure into a 5-inch-by-3-inch-by-4-inch rectangular prism that has a volume of 5×3×45 \times 3 \times 4 or 60 cubic inches, and a 3-inch-by-1-inch-by-4-inch rectangular prism that has a volume of 3×1×43 \times 1 \times 4 or 12 cubic inches. That’s 72 cubic inches total.

Problem 3

Find the volume of the figure. Explain or show your reasoning.

Two rectangular prisms, one in front of the other. 
Two rectangular prisms, one in front of the other. Front prism: 8 feet by 5 feet by 4 feet. Back prism: 4 feet by 6 feet by 4 feet. 

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Solution
304 cubic feet. Sample response: There are two rectangular prisms in the picture. The one in front is 4 ft by 5 ft by 8 ft. The volume is 160 cubic feet (4×5×84 \times 5 \times 8). The other rectangular prism has a volume of 144 cubic feet (4×6×64 \times6 \times 6). The total volume is 304 cubic feet.

Problem 4

This is a diagram of a bedroom. What is the volume of the bedroom? Explain or show your reasoning.

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Solution
992 cubic feet. Sample response: The room can be cut into a 4-foot by-4-foot-by-8 foot rectangular prism and a 12-foot-by-9-foot-by-8-foot rectangular prism. The volumes of those prisms are 128 cubic feet and 864 cubic feet, so the total volume is 992 cubic feet.

Problem 5

  1. Han says that the volume of this rectangular prism is 50 times as great as a 2-inch cube. Do you agree with Han? Explain or show your reasoning.

    Rectangular prism. 10 inch by 8 inch by 5 inch. 

  2. Han says that he can fit fifty 2-inch cubes in this rectangular prism. Do you agree with Han? Explain or show your reasoning.
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Solution

  1. Yes. Sample response: The rectangular prism has a volume of 5× 10× 85 \times 10 \times 8, which is 400 cubic inches. The 2-inch cube has a volume of 8 cubic inches, so the rectangular prism has a volume 50 times greater than the 2-inch cube.

  2. No. Sample response: The cubes have even measurements on all sides. There is no way to pack them so that they combine to give an odd measurement like the 5-inch height of the rectangular prism.

Problem 6

There are 2 common sizes of shipping boxes: 10 inches by 6 inches by 16 inches and 12 inches by 7 inches by 12 inches. Which size box would you choose to ship the books for your math class? Explain or show your reasoning.

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Solution
Sample responses: The side lengths of my book are about 11 inches by 9 inches by 38\frac38 inch. I can fit the books in the 10-inch-by-6-inch-by-16-inch box as long as I use the 10-inch-by-16-inch side as the base and lay the books flat in the box in a certain way. I can fit the books in the 12-inch-by-7-inch-by-12-inch box and as long as I use the 12-inch-by-12-inch side as the base, I can lay the books flat in the box in either direction, and I won’t have a lot of extra space.