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A rectangular prism and cube have equal volumes. The length of the rectangular prism is 12 centimeters (cm) and its width is
8cm. If each side of the cube is
12cm, then what is the height of the rectangular prism in centimeters?

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A rectangular prism and cube have equal volumes. The length of the rectangular prism is $12\,\text{cm}$ and its width is $8\,\text{cm}$ . If each side of the cube is $12\,\text{cm}$ , then what is the height of the rectangular prism in centimeters? $\newline$ $\square$

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Volume of cubes and rectangular prisms: word problems

Full solution

Q. A rectangular prism and cube have equal volumes. The length of the rectangular prism is

12\,\text{cm}

and its width is

8\,\text{cm}

. If each side of the cube is

12\,\text{cm}

, then what is the height of the rectangular prism in centimeters?

\newline

\square

Identify Volume of Cube: Identify the volume of the cube. $\newline$ Since the length of each side of the cube is $12 \, \text{cm}$ , the volume of the cube ( $V_{\text{cube}}$ ) can be calculated using the formula for the volume of a cube, which is $\text{side}^3$ . $\newline$ $V_{\text{cube}} = 12 \, \text{cm} \times 12 \, \text{cm} \times 12 \, \text{cm}$ $\newline$ $V_{\text{cube}} = 1728 \, \text{cubic centimeters}$
Express Volume of Prism: Express the volume of the rectangular prism. $\newline$ The volume of the rectangular prism $V_{\text{prism}}$ is equal to the volume of the cube since they have equal volumes. $\newline$ $V_{\text{prism}} = V_{\text{cube}}$ $\newline$ $V_{\text{prism}} = 1728$ cubic centimeters
Calculate Height of Prism: Calculate the height of the rectangular prism. $\newline$ The volume of a rectangular prism is calculated by multiplying its length, width, and height. We have the volume and the dimensions for length and width, so we can solve for the height $h$ . $\newline$ $V_{\text{prism}} = \text{length} \times \text{width} \times \text{height}$ $\newline$ $1728 \, \text{cm}^3 = 12 \, \text{cm} \times 8 \, \text{cm} \times h$
Solve for Height: Solve for the height. $\newline$ To find the height, divide both sides of the equation by the product of the length and width. $\newline$ $h = \frac{1728 \, \text{cm}^3}{(12 \, \text{cm} \times 8 \, \text{cm})}$ $\newline$ $h = \frac{1728 \, \text{cm}^3}{96 \, \text{cm}^2}$ $\newline$ $h = 18 \, \text{cm}$

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