In a right circular cone, the height is exactly twice the radius of its base. If the volume of the cone is cubic centimeters, what is the radius, in centimeters, of the base?

Solution available with step-by-step explanation

SAT Area & Volume Question 44 | Free SAT Prep | Aniko

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Question 44·Medium·Area and Volume

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In a right circular cone, the height is exactly twice the radius of its base. If the volume of the cone is $486\pi$ cubic centimeters, what is the radius, in centimeters, of the base?

Your answer

(Express the answer as an integer)

For cone volume questions, immediately write the standard formula $V = \frac{1}{3}\pi r^2 h$ and then substitute any given relationships (such as $h$ in terms of $r$ ) before plugging in the numerical volume. Simplify by canceling $\pi$ and combining like terms to reduce the equation to a single variable, then solve carefully—often it leads to a simple power equation like $r^3 = \text{number}$ , where you should recognize or quickly compute the cube root. Keeping algebra neat and isolating the variable step by step prevents arithmetic mistakes and saves time.

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Recall the volume formula

What is the formula for the volume of a cone in terms of its radius $r$ and height $h$ ? Write that formula first.

Use the height–radius relationship

You are told that the height is twice the radius. How can you write $h$ in terms of $r$ and substitute it into the cone volume formula?

Eliminate common factors

After substituting $h$ into the formula and setting the volume equal to $486\pi$ , notice that $\pi$ appears on both sides. What can you cancel to make the equation simpler?

Solve for $r$ step by step

Once you have an equation involving $r^3$ , solve for $r^3$ first, then take the cube root to get $r$ .

Desmos Guide

Enter the volume equation

In Desmos, type y = (2/3)*x^3 to represent the cone volume (with $h = 2r$ substituted) as a function of radius $x$ .

Find the radius

On a new line, type y = 486. Click on the intersection point. The x-coordinate is the radius of the base.

Verify with cube root

Alternatively, type 486^(1/3) to verify that $r^3 = 729$ gives $r = 9$ after working through the algebra.

Step-by-step Explanation

Write the cone volume formula

The volume $V$ of a right circular cone with base radius $r$ and height $h$ is

V = \frac{1}{3}\pi r^2 h.

You are given that $V = 486\pi$ cubic centimeters.

Use the relationship between height and radius

The problem states that the height is exactly twice the radius, so

h = 2r.

Substitute $h = 2r$ into the volume formula:

V = \frac{1}{3}\pi r^2 (2r).

This simplifies to

V = \frac{2}{3}\pi r^3.

Set up and simplify the equation for $r^3$

Now plug in the given volume $V = 486\pi$ :

486\pi = \frac{2}{3}\pi r^3.

Cancel $\pi$ from both sides:

486 = \frac{2}{3} r^3.

Multiply both sides by $\frac{3}{2}$ to solve for $r^3$ :

r^3 = 486 \cdot \frac{3}{2}.

Compute the product:

486 \cdot \frac{3}{2} = 729,

r^3 = 729.

Find the radius from $r^3 = 729$

To find $r$ , take the cube root of both sides:

r = \sqrt[3]{729}.

Since $9^3 = 9 \cdot 9 \cdot 9 = 81 \cdot 9 = 729$ , we get $r = 9$ .

Answer: $9$ .

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation