The radius of a sphere starts to increase at a rate of ` 0.1` cm/s . The rate of chane of a surface area of the sphere with time when radius is 10.

Answer

To solve the problem of finding the rate of change of the surface area of a sphere when the radius is increasing, we can follow these steps:

### Step 1: Understand the formula for the surface area of a sphere
The surface area \( S \) of a sphere is given by the formula:
\[ S = 4\pi r^2 \]
where \( r \) is the radius of the sphere.

### Step 2: Differentiate the surface area with respect to time
To find the rate of change of the surface area with respect to time, we need to differentiate \( S \) with respect to \( t \):
\[ \frac{dS}{dt} = \frac{dS}{dr} \cdot \frac{dr}{dt} \]
Using the chain rule, we first find \( \frac{dS}{dr} \).

### Step 3: Calculate \( \frac{dS}{dr} \)
Differentiating \( S = 4\pi r^2 \) with respect to \( r \):
\[ \frac{dS}{dr} = 8\pi r \]

### Step 4: Substitute known values
We know that the radius \( r \) is 10 cm and the rate of change of the radius \( \frac{dr}{dt} \) is 0.1 cm/s. Now we can substitute these values into the equation:
\[ \frac{dS}{dt} = \frac{dS}{dr} \cdot \frac{dr}{dt} = 8\pi r \cdot \frac{dr}{dt} \]
Substituting \( r = 10 \) and \( \frac{dr}{dt} = 0.1 \):
\[ \frac{dS}{dt} = 8\pi (10) \cdot (0.1) \]

### Step 5: Calculate \( \frac{dS}{dt} \)
Now we can calculate:
\[ \frac{dS}{dt} = 8\pi \cdot 10 \cdot 0.1 = 8\pi \text{ cm}^2/\text{s} \]

### Final Answer
Thus, the rate of change of the surface area of the sphere with respect to time when the radius is 10 cm is:
\[ \frac{dS}{dt} = 8\pi \text{ cm}^2/\text{s} \]

---