Find the rate of change of radius of a sphere when its radius is `4 cm` and when its volume is changing at the rate of `(4pi)/(625)m^(3)//sec`

Options

A. `1m//s`
B. `2m//s`
C. `pim//s`
D. `3m//s`

Explanation

To find the rate of change of the radius of a sphere when its radius is 4 cm and its volume is changing at a rate of \(\frac{4\pi}{625} \, m^3/s\), 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 time.
To find the rate of change of volume with respect to time, we differentiate both sides of the volume formula with respect to \(t\):
\[ \frac{dV}{dt} = \frac{d}{dt} \left( \frac{4}{3} \pi r^3 \right) \]
Using the chain rule, this becomes:
\[ \frac{dV}{dt} = \frac{4}{3} \pi (3r^2) \frac{dr}{dt} = 4\pi r^2 \frac{dr}{dt} \]

### Step 3: Substitute the known values.
We know that:
\[ \frac{dV}{dt} = \frac{4\pi}{625} \, m^3/s \]
and we need to find \(\frac{dr}{dt}\) when \(r = 4 \, cm = 0.04 \, m\) (since we need to convert cm to meters).

Substituting these values into the differentiated equation:
\[ \frac{4\pi}{625} = 4\pi (0.04)^2 \frac{dr}{dt} \]

### Step 4: Simplify the equation.
We can cancel \(4\pi\) from both sides:
\[ \frac{1}{625} = (0.04)^2 \frac{dr}{dt} \]
Calculating \((0.04)^2\):
\[ (0.04)^2 = 0.0016 \]
So we have:
\[ \frac{1}{625} = 0.0016 \frac{dr}{dt} \]

### Step 5: Solve for \(\frac{dr}{dt}\).
Rearranging the equation to solve for \(\frac{dr}{dt}\):
\[ \frac{dr}{dt} = \frac{1}{625 \times 0.0016} \]
Calculating \(625 \times 0.0016\):
\[ 625 \times 0.0016 = 1 \]
Thus:
\[ \frac{dr}{dt} = 1 \, m/s \]

### Final Answer:
The rate of change of the radius of the sphere when its radius is 4 cm is:
\[ \frac{dr}{dt} = 1 \, m/s \]