If a circular plate is heated uniformly, its area expands 3c times as fast as its radius, then the value of c when the radius is 6 units, is

Options

A. `4 pi`
B. `2 pi`
C. `6 pi`
D. `3 pi`

Answer

To solve the problem step by step, we will follow the reasoning presented in the video transcript:

### Step 1: Understand the relationship between area and radius
The area \( A \) of a circular plate is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the plate.

### Step 2: Differentiate the area with respect to time
To find how the area changes with respect to time, we differentiate both sides of the equation with respect to time \( t \):
\[ \frac{dA}{dt} = \frac{d}{dt}(\pi r^2) = \pi \cdot 2r \cdot \frac{dr}{dt} \]
This simplifies to:
\[ \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \]

### Step 3: Relate the change in area to the change in radius
According to the problem, the area expands \( 3c \) times as fast as its radius. This gives us the equation:
\[ \frac{dA}{dt} = 3c \frac{dr}{dt} \]

### Step 4: Set the two expressions for \(\frac{dA}{dt}\) equal to each other
From the two expressions we derived, we can set them equal:
\[ 2\pi r \frac{dr}{dt} = 3c \frac{dr}{dt} \]

### Step 5: Cancel \(\frac{dr}{dt}\) from both sides
Assuming that \(\frac{dr}{dt} \neq 0\), we can cancel it from both sides:
\[ 2\pi r = 3c \]

### Step 6: Solve for \( c \)
Now, we can solve for \( c \):
\[ c = \frac{2\pi r}{3} \]

### Step 7: Substitute the value of \( r \)
We need to find the value of \( c \) when the radius \( r = 6 \) units:
\[ c = \frac{2\pi \cdot 6}{3} \]
\[ c = \frac{12\pi}{3} = 4\pi \]

### Final Answer
The value of \( c \) when the radius is 6 units is:
\[ c = 4\pi \]