A circular metal plate is heated so that its radius increases at a rate of 0.1 mm per minute. Then the rate at which the plate's area is increasing when the radius is 50 cm, is

Options

A. `10pi` mm^2/minute
B. `100pi` mm^2/minute
C. `pi` mm^2/minute
D. `-pi` mm^2/minute

Answer

To solve the problem step by step, we will follow the mathematical principles of derivatives and rates of change.

### Step 1: Understand the Given Information
We are given:
- The rate of increase of the radius of the circular plate: \( \frac{dr}{dt} = 0.1 \text{ mm/min} \)
- We need to find the rate of change of the area when the radius \( r = 50 \text{ cm} \).

### Step 2: Convert Units
Since the radius is given in centimeters and the rate of increase is in millimeters, we should convert the radius into millimeters for consistency:
\[ r = 50 \text{ cm} = 50 \times 10 \text{ mm} = 500 \text{ mm} \]

### Step 3: Write the Formula for Area
The area \( A \) of a circular plate is given by the formula:
\[ A = \pi r^2 \]

### Step 4: Differentiate the Area with Respect to Time
To find the rate of change of the area with respect to time, we differentiate both sides of the area formula with respect to \( t \):
\[ \frac{dA}{dt} = \frac{d}{dt}(\pi r^2) = \pi \cdot 2r \cdot \frac{dr}{dt} \]
This simplifies to:
\[ \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \]

### Step 5: Substitute the Known Values
Now, we substitute \( r = 500 \text{ mm} \) and \( \frac{dr}{dt} = 0.1 \text{ mm/min} \) into the equation:
\[ \frac{dA}{dt} = 2\pi (500) \left(0.1\right) \]

### Step 6: Calculate the Rate of Change of Area
Calculating this gives:
\[ \frac{dA}{dt} = 2\pi \cdot 500 \cdot 0.1 = 100\pi \text{ mm}^2/\text{min} \]

### Final Answer
Thus, the rate at which the area of the plate is increasing when the radius is 50 cm is:
\[ \frac{dA}{dt} = 100\pi \text{ mm}^2/\text{min} \]