The rate of change of volume of a sphere with respect to its radius when radius is 1 unit is :

Options

A. `4pi`
B. `2pi`
C. `pi`
D. `(pi)/(2).`

Answer

To find the rate of change of the volume of a sphere with respect to its radius when the radius is 1 unit, we can follow these steps:

### Step 1: Write the formula for the volume of a sphere.
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]

### Step 2: Differentiate the volume with respect to the radius.
To find the rate of change of volume with respect to the radius, we need to differentiate \( V \) with respect to \( r \):
\[ \frac{dV}{dr} = \frac{d}{dr} \left( \frac{4}{3} \pi r^3 \right) \]
Using the power rule of differentiation, we get:
\[ \frac{dV}{dr} = \frac{4}{3} \pi \cdot 3r^2 \]
The \( 3 \) in the numerator and denominator cancels out:
\[ \frac{dV}{dr} = 4 \pi r^2 \]

### Step 3: Substitute the radius value.
We need to find the rate of change when the radius \( r \) is 1 unit:
\[ \frac{dV}{dr} \bigg|_{r=1} = 4 \pi (1)^2 \]
Calculating this gives:
\[ \frac{dV}{dr} \bigg|_{r=1} = 4 \pi \]

### Conclusion
Thus, the rate of change of volume of the sphere with respect to its radius when the radius is 1 unit is:
\[ \boxed{4\pi} \]

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