The rate of change of surface area of a sphere of radius r when the radius is increasing at the rate of 2 cm/sec is proportional to

Options

A. `(1)/(r^(2))`
B. `(1)/(r )`
C. `r^(2)`
D. r

Explanation

To solve the problem, we need to find the rate of change of the surface area of a sphere when the radius is increasing at a specific rate. Let's go through the steps systematically.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- The radius \( r \) of the sphere is increasing at a rate of \( \frac{dr}{dt} = 2 \) cm/sec.

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

3. **Differentiate the Surface Area with Respect to Time**:
- To find the rate of change of surface area with respect to time, we differentiate \( S \) with respect to \( t \):
\[ \frac{dS}{dt} = \frac{d}{dt}(4\pi r^2) \]
- Using the chain rule, we get:
\[ \frac{dS}{dt} = 4\pi \cdot 2r \cdot \frac{dr}{dt} \]
- This simplifies to:
\[ \frac{dS}{dt} = 8\pi r \cdot \frac{dr}{dt} \]

4. **Substitute the Rate of Change of Radius**:
- Now, substitute \( \frac{dr}{dt} = 2 \) cm/sec into the equation:
\[ \frac{dS}{dt} = 8\pi r \cdot 2 = 16\pi r \]

5. **Determine the Proportionality**:
- From the expression \( \frac{dS}{dt} = 16\pi r \), we can see that the rate of change of surface area \( \frac{dS}{dt} \) is directly proportional to \( r \).
- Therefore, we conclude that:
\[ \frac{dS}{dt} \propto r \]

### Final Answer:
The rate of change of surface area of a sphere is proportional to \( r \).

---