Right rectangular prism is similar to right rectangular prism . The surface area of prism is , and the volume of prism is . The surface area of prism is . What is the volume, in , of prism ?

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SAT Area & Volume Question 36 | Free SAT Prep | Aniko

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Question 36·Hard·Area and Volume

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Right rectangular prism $P$ is similar to right rectangular prism $Q$ . The surface area of prism $P$ is $150\text{ cm}^2$ , and the volume of prism $P$ is $125\text{ cm}^3$ . The surface area of prism $Q$ is $2,400\text{ cm}^2$ . What is the volume, in $\text{cm}^3$ , of prism $Q$ ?

Your answer

(Express the answer as an integer)

For similarity problems with 3D figures, avoid trying to find individual side lengths; instead, use scale factors. First divide the given surface areas to find the area ratio, recognize that equals the square of the linear scale factor, and take the square root to get the linear factor. Then cube this linear factor to get the volume scale factor and multiply the known volume by this factor. This approach is quick, systematic, and reduces arithmetic errors on the SAT.

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Connect similarity to area and volume

When two 3D figures are similar, how do their surface areas and volumes change compared to the linear scale factor between them?

Start with the surface areas

Compute the ratio of the surface areas of prism Q to prism P: $2400 \div 150$ . What does this ratio represent in terms of the scale factor?

From surface area ratio to volume ratio

Once you know that the surface area ratio equals $k^2$ , find $k$ . Then use $k^3$ to determine how the volume of prism Q compares to the volume of prism P.

Apply the volume scale factor

Multiply the volume of prism P, 125 cm³, by your volume scale factor to get the volume of prism Q.

Desmos Guide

Compute the surface area ratio

In Desmos, enter: area_ratio = 2400/150. This gives you the ratio of the surface areas of prism Q to prism P.

Find the linear and volume scale factors

In a new line, enter: k = sqrt(area_ratio) to get the linear scale factor, then enter: volume_scale = k^3 to get the volume scale factor from P to Q.

Calculate the volume of prism Q

Finally, enter: volume_Q = 125*volume_scale. The value shown for volume_Q is the volume of prism Q in cubic centimeters.

Step-by-step Explanation

Use similarity relationships for 3D figures

For similar 3D solids:

If the linear scale factor (ratio of corresponding lengths) is $k$ , then the ratio of surface areas is $k^2$ and the ratio of volumes is $k^3$ .

So our plan is:

Use the surface areas to find $k^2$ .
Find $k$ .
Use $k^3$ to scale the volume from prism P to prism Q.

Find the surface area ratio

We are given:

Surface area of prism P: $150\text{ cm}^2$
Surface area of prism Q: $2400\text{ cm}^2$

So the ratio of surface areas (Q to P) is

\frac{\text{SA}_Q}{\text{SA}_P} = \frac{2400}{150} = 16.

This means $k^2 = 16$ , where $k$ is the linear scale factor from P to Q.

Find the linear and volume scale factors

From $k^2 = 16$ we get the linear scale factor:

k = \sqrt{16} = 4.

The volume scale factor is then

k^3 = 4^3 = 64.

So the volume of prism Q is 64 times the volume of prism P.

Scale the volume of prism P to get the volume of prism Q

We are given the volume of prism P is $125\text{ cm}^3$ .

Use the volume scale factor 64:

\text{Volume of Q} = 125 \times 64.

Compute this:

$125 \times 8 = 1000$
Then $1000 \times 8 = 8000$ so $125 \times 64 = 8000$ .

Therefore, the volume of prism Q is $8000\text{ cm}^3$ .

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation