The base of a right circular cylinder has a diameter of 10 centimeters. The volume of the cylinder is cubic centimeters. What is the height, in centimeters, of the cylinder?

Solution available with step-by-step explanation

SAT Area & Volume Question 41 | Free SAT Prep | Aniko

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Question 41·Medium·Area and Volume

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The base of a right circular cylinder has a diameter of 10 centimeters. The volume of the cylinder is $1{,}250\pi$ cubic centimeters. What is the height, in centimeters, of the cylinder?

For cylinder volume questions, immediately write down $V = \pi r^2 h$ and convert any given diameter to radius by dividing by 2. Compute $r^2$ , plug into the formula, and set the expression equal to the given volume. To save time and avoid mistakes, cancel $\pi$ on both sides as early as possible, then divide to solve for $h$ , watching carefully that you squared the radius and did not confuse the radius with the diameter or the base area with the height.

Hints

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Recall the correct volume formula

What is the formula for the volume of a right circular cylinder in terms of the radius $r$ and height $h$ ?

Relate diameter and radius

The problem gives you the diameter of the base. How do you find the radius from the diameter?

Connect base area and volume

First find the area of the circular base using the radius. Then, use the idea that volume equals base area times height to write an equation and solve for the height.

Solve the equation carefully

Once you have an equation of the form $1{,}250\pi = (\text{something}) \cdot h$ , divide both sides by that "something" (including any $\pi$ ) to isolate $h$ .

Desmos Guide

Compute the height directly

In a new expression line in Desmos, type (1250*pi)/(pi*(10/2)^2) and let Desmos evaluate it; the value it outputs is the height of the cylinder in centimeters.

Step-by-step Explanation

Use the cylinder volume formula

For a right circular cylinder, the volume formula is

V = \pi r^2 h

where $r$ is the radius of the base and $h$ is the height.

Find the radius and base area

The diameter of the base is 10 cm, so the radius is half of that:

r = \frac{10}{2} = 5

The area of the circular base is

A = \pi r^2 = \pi (5)^2 = 25\pi.

Set up an equation for the volume

Volume equals base area times height:

V = (\text{base area}) \cdot h = 25\pi \cdot h.

We are told the volume is $1{,}250\pi$ , so

1{,}250\pi = 25\pi h.

Solve for the height

Divide both sides of the equation by $25\pi$ to solve for $h$ :

\begin{aligned} 1{,}250\pi &= 25\pi h \\ \frac{1{,}250\pi}{25\pi} &= h \\ h &= 50 \end{aligned}

So the height of the cylinder is 50 centimeters, which corresponds to choice (C).

Strategy

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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