The volume of a sphere is changing @ 100 cc/min. The rate at which the surface area of the sphere is changing when the radius of the sphere = 10 cm, is

Options

A. `30 cm^(2)// min`
B. `20 cm^(2)//min`
C. `20 pi cm^(2)//min`
D. `30 pi cm^(2) //min`

Answer

To solve the problem, we need to find the rate at which the surface area of a sphere is changing, given that the volume of the sphere is changing at a rate of 100 cc/min when the radius is 10 cm.

### Step-by-step Solution:

1. **Understand the given information**:
- The rate of change of volume (\( \frac{dV}{dt} \)) is given as 100 cc/min.
- The radius (\( r \)) at the moment we are interested in is 10 cm.

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

3. **Differentiate the volume with respect to time**:
To find the rate of change of volume with respect to time, we differentiate \( V \):
\[ \frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt} \]

4. **Substitute the known values**:
We know \( \frac{dV}{dt} = 100 \) and \( r = 10 \) cm. Substitute these values into the differentiated equation:
\[ 100 = 4 \pi (10^2) \frac{dr}{dt} \]
\[ 100 = 4 \pi (100) \frac{dr}{dt} \]
\[ 100 = 400 \pi \frac{dr}{dt} \]

5. **Solve for \( \frac{dr}{dt} \)**:
Rearranging the equation gives:
\[ \frac{dr}{dt} = \frac{100}{400 \pi} = \frac{1}{4 \pi} \text{ cm/min} \]

6. **Surface area of a sphere formula**:
The surface area \( S \) of a sphere is given by the formula:
\[ S = 4 \pi r^2 \]

7. **Differentiate the surface area with respect to time**:
To find the rate of change of surface area with respect to time, we differentiate \( S \):
\[ \frac{dS}{dt} = 8 \pi r \frac{dr}{dt} \]

8. **Substitute the known values**:
Substitute \( r = 10 \) cm and \( \frac{dr}{dt} = \frac{1}{4 \pi} \):
\[ \frac{dS}{dt} = 8 \pi (10) \left(\frac{1}{4 \pi}\right) \]
\[ \frac{dS}{dt} = 80 \left(\frac{1}{4}\right) = 20 \text{ cm}^2/\text{min} \]

### Final Answer:
The rate at which the surface area of the sphere is changing when the radius is 10 cm is \( 20 \text{ cm}^2/\text{min} \).