Section B Practice Problems

Problem 1

Andre and Clare used different strategies to find the volume of this rectangular prism.

Andre says the volume of this rectangular prism is $8 \times 24$ cubic units. Is Andre correct? Explain or show your reasoning.
Clare says the volume of the rectangular prism is $6 \times 32$ cubic units. Is this correct? Explain or show your reasoning.

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Solution

Sample responses:

There are 8 layers of $6 \times 4$ unit cubes if the 6-unit-by-4-unit face is used as a base.
There are 6 layers of $8 \times 4$ unit cubes if the 8-unit-by-4-unit face is used as a base.

Problem 2

Which expressions represent the volume of this rectangular prism in cubic units?

Select all that apply.

Rectangular prism. 6 units by 3 units by 4 units.

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Solution

A,D,E

Problem 3

A box of milk measures 4 centimeters by 10 centimeters by 30 centimeters. What is its volume in cubic centimeters? Explain or show your reasoning.

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Solution

1,200 cubic cm. Sample response:

4 \times 10 \times 30

40 \times 30

or 1,200 cubic cm.

Problem 4

A sugar cube has a volume of about 1 cubic centimeter. Estimate the size of a box you would need to hold 1,000,000 sugar cubes.

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Solution

Sample response:

1,000,000 = 100 \times 100 \times 100

, so a cube that is 100 cm on each side would be big enough to hold 1,000,000 sugar cubes. This is 1 meter by 1 meter by 1 meter, so it is a very big box.

Problem 5

Find some objects around the school or your home. What units would you use to measure their volumes? Choose one of your objects and estimate its volume.

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Solution

Sample response 1: My lunch box is 9 inches wide, 6 inches in length, and 2 inches high. The volume is $9 \times 6 \times 2$ cubic inches or 108 cubic inches.

Sample response 2: My math book is 28 cm long, 22 cm wide, and 1 cm deep. The volume is $28 \times 22 \times 1$ or 616 cubic centimeters.

Problem 6

An object has a volume of 36 cubic inches. A box has side lengths 1 foot by 3 inches by 4 inches.

What is the least number of these objects that can fit within the box? Explain your reasoning.
What is the greatest number of these objects that can fit within the box? Explain your reasoning.

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Solution

0 objects. Sample response: If the object is 36 inches by 1 inch by 1 inch, a tall tower of unit inch cubes, none would fit in the box.
4 objects. Sample response: If the object is 12 inches by 3 inches by 1 inch, then 4 of them would fit in the box, with no extra space.

Problem 7

A container has a volume of 120 cubic inches.

What could be the length, the width, and the height of the container?
Can one of the side lengths be 9 inches? Explain or show your reasoning.

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Solution

Sample responses: 4 inches by 5 inches by 6 inches, 3 inches by 8 inches by 5 inches, 10 inches by 3 inches by 4 inches (any 3 whole-number side lengths whose product is 120)
Sample responses:
- No. 9 is not a factor of 120, so there are no whole numbers for the length, the width, and the height, with a product of 120, if one of those numbers is 9.
- Yes. If fractional lengths are possible, then the side lengths could be 9 inches by 5 inches by $\frac{8}{3}$ inches.