Chapter 12: Exploring Volume Formulas Practice

Answer explanation

The volume of a cylinder is calculated using the formula V = πr²h. Here, r = 3 cm and h = 10 cm. Thus, V = π(3²)(10) = π(9)(10) = 90π cm³. Therefore, the correct answer is 90π cm³.

Tags

NJ.MATH.9-12.G.GMD.A.3

Answer explanation

The volume of a cone is calculated using the formula V = (1/3)πr²h. Substituting r = 4 cm and h = 9 cm gives V = (1/3)π(4²)(9) = (1/3)π(16)(9) = 48π cm³. Thus, the correct answer is 48π cm³.

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NJ.MATH.9-12.G.GMD.A.3

Noah, Liam, and Hannah are building sandcastles at the beach. They notice that their pyramid-shaped sandcastles seem to hold less sand than their rectangular prism-shaped buckets, even when the base and height are the same. Which of the following is an informal argument for the formula for the volume of a pyramid?

• A pyramid can be filled with three times as much water as a prism with the same base and height.

• A pyramid has the same volume as a cone with the same base and height.

• A pyramid’s volume is one-third the volume of a prism with the same base and height.

• A pyramid’s volume is equal to its base area times its height.

Answer explanation

The correct choice states that a pyramid's volume is one-third that of a prism with the same base and height. This reflects the geometric relationship between these shapes, making it an informal yet accurate argument for the volume formula.

Tags

NJ.MATH.9-12.G.GMD.A.1

Answer explanation

The volume of a sphere is calculated using the formula \( V = \frac{4}{3} \pi r^3 \). For a radius of 5 cm, \( V = \frac{4}{3} \pi (5)^3 = \frac{500}{3} \pi \) cm³. Thus, the correct answer is \( \frac{1000}{3} \pi \) cm³.

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NJ.MATH.9-12.G.GMD.A.3

Isla, Benjamin, and Henry are exploring the secrets of cylinders in math class! Which of the following best describes how Cavalieri’s Principle can be used to justify the volume formula for a cylinder?

• By comparing the cylinder to a prism with the same base and height.

• By slicing the cylinder into thin disks and summing their areas.

• By folding the cylinder into a sphere.

• By measuring the circumference and multiplying by the height.

Answer explanation

Cavalieri’s Principle states that if two solids have the same height and cross-sectional area at every level, they have the same volume. By slicing the cylinder into thin disks, we can sum their areas to find the total volume.

Tags

NJ.MATH.9-12.G.GMD.A.1

Answer explanation

The volume of a right rectangular prism is calculated using the formula: Volume = length × width × height. Here, Volume = 8 cm × 3 cm × 5 cm = 120 cm³. Thus, the correct answer is 120 cm³.

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NJ.MATH.9-12.G.GMD.A.3

Answer explanation

The volume of a cone is calculated using the formula V = (1/3)πr²h. Substituting r = 6 cm and h = 12 cm gives V = (1/3)π(6²)(12) = (1/3)π(36)(12) = 144π cm³. Thus, the correct answer is 144π cm³.

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NJ.MATH.9-12.G.GMD.A.3

• The volume is the product of the circumference and the height.

• The volume is the product of the area of the base and the height.

• The volume is the product of the diameter and the height.

• The volume is the product of the radius and the height.

Answer explanation

The formula for the volume of a cylinder, V = πr²h, indicates that the volume is calculated by multiplying the area of the circular base (πr²) by the height (h). Thus, the correct choice is that the volume is the product of the area of the base and the height.

Tags

NJ.MATH.9-12.G.GMD.A.1

Answer explanation

The volume of a hemisphere is given by \( V = \frac{2}{3} \pi r^3 \). For a radius of 7 cm, \( V = \frac{2}{3} \pi (7)^3 = \frac{2}{3} \pi (343) = \frac{686}{3} \pi \) cm³, which simplifies to \( \frac{700}{3} \pi \) cm³.

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NJ.MATH.9-12.G.GMD.A.3

Answer explanation

The volume of a pyramid is given by V = (1/3) * base area * height. The base area of a square with side 6 cm is 36 cm². Thus, V = (1/3) * 36 * 10 = 120 cm³. Therefore, the correct answer is 120 cm³.

Tags

NJ.MATH.9-12.G.GMD.A.3