Question

The radius of a sphere (in cm) whose volume is \(36 \pi \text{ cm}^{3}\), is :

• A. 3
• B. \(3\sqrt{3}\)
• C. \(3^{\frac{2}{3}}\)
• D. \(3^{\frac{1}{3}}\)

Hint:

Keep an eye out for perfect cubes like 1, 8, 27, 64 in these problems. Once you reach \(r^{3} = 27\), it is easy to see the answer is 3.

Answer 1

The Correct Option is A

To find the radius of a sphere given its volume, we start with the formula for the volume of a sphere:

\(V = \frac{4}{3} \pi r^3\)

We are given that the volume \(V = 36 \pi \text{ cm}^3\). Let's equate this to the sphere volume formula and solve for the radius \(r\):

\(\frac{4}{3} \pi r^3 = 36 \pi\)

Firstly, divide both sides by \(\pi\) to eliminate it:

\(\frac{4}{3} r^3 = 36\)

Next, multiply both sides by \(\frac{3}{4}\) to solve for \(r^3\):

\(r^3 = 36 \times \frac{3}{4}\)

Simplify the right side:

\(r^3 = 27\)

To find \(r\), take the cube root of both sides:

\(r = \sqrt[3]{27}\)

This simplifies to:

\(r = 3\)

Thus, the radius of the sphere is 3 cm. Let's verify this solution by checking if the other options could be the radius:

• \(r = 3\sqrt{3}\) — Substituting this into the volume formula would give a volume greater than 36\(\pi\).
• \(r = 3^{\frac{2}{3}}\) — This would give a smaller volume than required.
• \(r = 3^{\frac{1}{3}}\) — This option would also give a smaller volume.

Therefore, the only correct and viable option is 3 cm.