Question

If the radius of a sphere is 2r, then its volume will be :

• A. \(\frac{4}{3}\pi r^3\)
• B. \(\frac{32\pi r^3}{3}\)
• C. \(\frac{2\pi r^3}{3}\)
• D. \(4\pi r^3\)

Hint:

When the radius of a 3D object is multiplied by a factor of \(k\), its volume is multiplied by \(k^3\).
Here, the radius is doubled (\(k = 2\)), so the volume increases by a factor of \(2^3 = 8\).
Multiplying the standard volume \(\frac{4}{3}\pi r^3\) by 8 gives \(\frac{32}{3}\pi r^3\) instantly.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question is from Mensuration, dealing with the volume of a sphere.
We are given the radius of a sphere in terms of a variable \(r\), and we need to calculate its volume.

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 substitute the given radius \(R = 2r\) into this formula.

Step 3: Detailed Explanation:
Let the radius of the sphere be:
\[ R = 2r \]
Using the standard volume formula:
\[ V = \frac{4}{3}\pi R^3 \]
Substitute \(R = 2r\):
\[ V = \frac{4}{3}\pi (2r)^3 \]
First, calculate the cube of \(2r\):
\[ (2r)^3 = 2^3 \times r^3 = 8r^3 \]
Now, substitute this back into the expression:
\[ V = \frac{4}{3}\pi \times 8r^3 \]
Multiply the constants:
\[ V = \frac{4 \times 8}{3} \pi r^3 \]
\[ V = \frac{32}{3}\pi r^3 \]

Step 4: Final Answer:
The volume of the sphere is \(\frac{32\pi r^3}{3}\).