The rate of change of volume of a sphere is equal to the rate of change of its radius, then its radius is equal to

Options

A. 1
B. 2
C. 0.5
D. none of these

Explanation

To solve the problem, we need to find the radius of a sphere given that the rate of change of its volume is equal to the rate of change of its radius.

### Step-by-Step Solution:

1. **Understand the Volume of a Sphere**:
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]

2. **Differentiate the Volume with Respect to Time**:
To find the rate of change of volume with respect to time, we differentiate \( V \) with respect to \( t \):
\[ \frac{dV}{dt} = \frac{d}{dt}\left(\frac{4}{3} \pi r^3\right) \]
Using the chain rule, we get:
\[ \frac{dV}{dt} = \frac{4}{3} \pi \cdot 3r^2 \frac{dr}{dt} = 4\pi r^2 \frac{dr}{dt} \]

3. **Set the Rate of Change of Volume Equal to the Rate of Change of Radius**:
According to the problem, the rate of change of volume is equal to the rate of change of radius:
\[ \frac{dV}{dt} = \frac{dr}{dt} \]
Therefore, we can set the two expressions equal to each other:
\[ 4\pi r^2 \frac{dr}{dt} = \frac{dr}{dt} \]

4. **Cancel \(\frac{dr}{dt}\)**:
Assuming \(\frac{dr}{dt} \neq 0\), we can divide both sides by \(\frac{dr}{dt}\):
\[ 4\pi r^2 = 1 \]

5. **Solve for \( r^2 \)**:
Rearranging the equation gives us:
\[ r^2 = \frac{1}{4\pi} \]

6. **Find \( r \)**:
Taking the square root of both sides, we find:
\[ r = \frac{1}{2\sqrt{\pi}} \]

### Final Answer:
The radius \( r \) is equal to \( \frac{1}{2\sqrt{\pi}} \).

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