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:

When dealing with $\pi$ in solid geometry formulas, always cancel $\pi$ first before doing any cross-multiplication.
Recognizing perfect cubes like $27 = 3^3$, $64 = 4^3$, and $125 = 5^3$ helps to speed up calculations.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given the volume of a sphere as $36\pi \text{ cm}^3$.
We need to calculate the radius of this sphere in centimeters.

Step 2: Key Formula or Approach:
The volume $V$ of a sphere of radius $r$ is given by the formula:
\[ V = \frac{4}{3}\pi r^3 \]
We will equate this formula to the given volume and solve for $r$.

Step 3: Detailed Explanation:

• Write down the given volume:
\[ V = 36\pi \text{ cm}^3 \]

• Substitute the volume formula:
\[ \frac{4}{3}\pi r^3 = 36\pi \]

• Cancel the common term $\pi$ from both sides of the equation:
\[ \frac{4}{3} r^3 = 36 \]

• Multiply both sides by 3 to clear the denominator:
\[ 4r^3 = 36 \times 3 \]
\[ 4r^3 = 108 \]

• Divide both sides by 4 to isolate $r^3$:
\[ r^3 = \frac{108}{4} \]
\[ r^3 = 27 \]

• Take the cube root of both sides to find $r$:
\[ r = \sqrt[3]{27} \]
Since $3 \times 3 \times 3 = 27$:
\[ r = 3 \text{ cm} \]


Step 4: Final Answer:
The radius of the sphere is 3 cm.
Hence, option (A) is correct.