Super-Hard SAT Math Question 49 | Free SAT Prep | Aniko

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Question 49·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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Two closed right circular cylinders, Cylinder A and Cylinder B, are similar.

The total surface area of Cylinder A is $500\pi$ square centimeters, and the total surface area of Cylinder B is $1,280\pi$ square centimeters. The volume of Cylinder A is $250\pi$ cubic centimeters.

Which choice is the volume of Cylinder B, in cubic centimeters?

When a problem says two 3D figures are similar, immediately think “length $\to k$ , area $\to k^2$ , volume $\to k^3$ .” Use the given surface areas to find $k$ by taking a square root, then use $k^3$ to scale the volume. Cancel common factors (like $\pi$ ) early to keep the arithmetic simple.

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Use similarity scaling rules

For similar solids, areas scale with the square of the scale factor and volumes scale with the cube of the scale factor.

Start from the surface areas

Compute $\frac{\text{SA}_B}{\text{SA}_A}$ . That ratio equals $k^2$ for some linear scale factor $k$ .

Move from $k^2$ to volume

After you find $k$ , multiply the given volume by $k^3$ (not by $k$ or $k^2$ ).

Desmos Guide

Compute the surface-area ratio

Enter 1280/500 to get the ratio $\frac{\text{SA}_B}{\text{SA}_A}$ (the $\pi$ cancels).

Find the linear scale factor

Enter sqrt(1280/500) to compute $k$ .

Apply volume scaling

Enter 250*(sqrt(1280/500))^3 to compute the volume multiplier applied to 250. Then attach $\pi$ and match the result to one of the answer choices.

Step-by-step Explanation

Find the surface area scale factor

Because the cylinders are similar, the ratio of their total surface areas equals $k^2$ , where $k$ is the linear scale factor from Cylinder A to Cylinder B.

\frac{\text{SA}_B}{\text{SA}_A} = \frac{1{,}280\pi}{500\pi} = \frac{1280}{500} = \frac{64}{25}

So $k^2=\frac{64}{25}$ .

Convert to the linear scale factor

Take the square root to get $k$ :

k=\sqrt{\frac{64}{25}}=\frac{8}{5}

Scale the volume

Volume scales by $k^3$ , so

\text{V}_B = \text{V}_A \cdot k^3 = 250\pi \left(\frac{8}{5}\right)^3 = 250\pi \cdot \frac{512}{125} = 2\cdot 512\pi = 1024\pi

So the volume of Cylinder B is $1024\pi$ cubic centimeters.

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Step-by-step Explanation

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Step-by-step Explanation