SAT Area & Volume Question 42 | Free SAT Prep | Aniko

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Question 42·Hard·Area and Volume

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A solid is created by attaching a hemisphere to each end of a right circular cylinder, forming a shape that resembles a capsule. Each hemisphere and the cylinder share the same radius of $6\text{ cm}$ , and the cylinder’s height (not including the hemispheres) is $10\text{ cm}$ . What is the volume, in cubic centimeters, of the entire solid?

For composite solids on the SAT, first break the shape into familiar pieces (cylinders, cones, spheres, prisms). Write the standard volume formula for each piece, plug in the correct dimensions (being careful with radius vs. diameter and which measurements apply to which shape), and compute each volume separately. Finally, add or subtract these volumes as the problem describes; in capsule shapes, remember that two hemispheres of the same radius form one full sphere.

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Identify the component shapes

Ignore the complicated overall shape for a moment. What simple 3D shapes make up this solid, and how many of each are there?

Relate hemispheres to a sphere

You have two hemispheres of the same radius. How do their combined volumes compare to the volume of one full sphere with that radius?

Recall the volume formulas

Write down the volume formulas for a cylinder and a sphere. Then substitute $r = 6$ into each, and use $h = 10$ only for the cylinder.

Combine your results

After you find the cylinder’s volume and the total volume of the two hemispheres, how do you combine those numbers to get the volume of the entire solid?

Desmos Guide

Use Desmos to confirm the total volume expression

In Desmos, type the full volume calculation as one expression: pi*6^2*10 + (4/3)*pi*6^3. The value Desmos outputs is the total volume of the solid in cubic centimeters. Compare this result with each answer choice (by converting the choices to decimals if needed) and select the one that matches.

Step-by-step Explanation

Break the solid into basic shapes

The “capsule” shape is made of:

A right circular cylinder of radius $6$ cm and height $10$ cm.
Two hemispheres, each of radius $6$ cm, attached to the ends of the cylinder.

Two hemispheres of the same radius together form one full sphere of radius $6$ cm.

Compute the volume of the cylinder

Use the cylinder volume formula $V = \pi r^2 h$ .

Here, $r = 6$ and $h = 10$ :

$V_{\text{cyl}} = \pi (6)^2(10) = \pi(36)(10) = 360\pi$ .

So the cylinder contributes $360\pi$ cubic centimeters.

Compute the volume of the two hemispheres (one sphere)

Two hemispheres make one full sphere of radius $6$ cm.

The volume of a sphere is $V = \frac{4}{3}\pi r^3$ .

For $r = 6$ :

V_{\text{sphere}} = \frac{4}{3}\pi(6)^3 = \frac{4}{3}\pi(216) = 288\pi

So the two hemispheres together contribute $288\pi$ cubic centimeters.

Add the volumes to get the total

Add the volume of the cylinder and the volume of the sphere:

V_{\text{total}} = V_{\text{cyl}} + V_{\text{sphere}} = 360\pi + 288\pi = 648\pi.

So the volume of the entire solid is $648\pi$ cubic centimeters, which corresponds to answer choice C.

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation