Question 189·Hard·Geometry and Trigonometry

If the diameter of sphere B is 50% greater than the diameter of sphere A, by what factor is the radius of sphere B larger than the radius of sphere A?

When the radius is multiplied by some factor $k$, how does the volume change? Express the new volume in terms of $k$ and the original volume.

You know the original volume is $288\pi$ and you know the factor by which volume changes. Multiply them to get the new volume, then match it to the closest answer choice.

In the expression line, type (1.5)^3 * 288 * pi and press Enter. The numerical result shown is the volume of sphere B in cubic centimeters; match this value (in terms of $\pi$) to one of the answer choices.

The volume $V$ of a sphere with radius $r$ is given by

This means volume is proportional to the cube of the radius.

Sphere B has a diameter that is 50% greater than sphere A.

• A 50% increase means multiplying by $1.5$ (since $100\% + 50\% = 150\% = 1.5$).
• So the diameter of sphere B is $1.5$ times the diameter of sphere A.
• Radius is half of diameter, so the radius of sphere B is also $1.5$ times the radius of sphere A.

When you scale the radius of a sphere by a factor of $k$, the volume is scaled by $k^3$ (because of the $r^3$ in the formula).

Here, the radius of sphere B is $1.5$ times the radius of sphere A, so the volume of sphere B is

Compute $(1.5)^3$:

Sphere A has volume $V_A = 288\pi$ cubic centimeters.

Use the volume scale factor $\frac{27}{8}$:

First simplify $288 \div 8 = 36$, so

Now multiply:

So the volume of sphere B is $972\pi$ cubic centimeters, which corresponds to choice D.