The table gives the volume and surface area of two similar right rectangular prisms, where is a constant. | | Volume (cubic centimeters) | Surface Area (square centimeters) | |---|---|---| | Prism | | | | Prism | | | Which choice is the value of ?

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Super-Hard SAT Math Question 166 | Free SAT Prep | Aniko

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Question 166·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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The table gives the volume and surface area of two similar right rectangular prisms, where $k$ is a constant.

	Volume (cubic centimeters)	Surface Area (square centimeters)
Prism $A$	$5{,}832$	$1{,}458$
Prism $B$	$1{,}259{,}712$	$k$

Which choice is the value of $k$ ?

When two 3D solids are similar, avoid trying to reconstruct dimensions. Instead, use scaling laws: if the linear scale factor is $s$ , then volumes scale by $s^3$ and surface areas scale by $s^2$ . So compute the volume ratio first, take its cube root to get $s$ , then multiply the given surface area by $s^2$ .

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Use similarity scaling

For similar 3D solids, volume scales with the cube of the scale factor, while surface area scales with the square.

Start with the volumes

Compute the ratio $\frac{\text{Volume of }B}{\text{Volume of }A}$ and look for it to be a perfect cube.

Then connect to surface area

Take the cube root to get the linear scale factor, then square it to get how surface area changes.

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Compute the volume ratio

Enter 1259712/5832 and note the value (this is the volume scale factor).

Find the linear scale factor

Enter (1259712/5832)^(1/3) to compute the cube root of the volume ratio.

Scale the surface area

Enter 1458*((1259712/5832)^(1/3))^2 and match the result to one of the answer choices.

Step-by-step Explanation

Find the volume scale factor

Because the prisms are similar, the ratio of their volumes equals the cube of the linear scale factor.

Compute the volume ratio:

\frac{1{,}259{,}712}{5{,}832} = 216

Convert to the linear scale factor

If the linear scale factor is $s$ , then $s^3=216$ .

So $s=\sqrt[3]{216}=6$ .

Scale the surface area

Surface area scales by the square of the linear scale factor, so the surface area scale factor is $s^2=6^2=36$ .

Then

k = 1{,}458\cdot 36 = 52{,}488

So the correct choice is $52{,}488$ .

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