Question

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

• A. \( 4\pi \)
• B. \( 2\pi \)
• C. \( 6\pi \)
• D. \( 3\pi \)
• E. \( 8\pi \)

Hint:

In rate problems, always: 1. Write the formula first 2. Differentiate with respect to time 3. Substitute the given rate relation 4. Solve systematically

Answer 1

The Correct Option is A

Concept:
• Area of a circle: \[ A = \pi r^2 \]
• When quantities change with time, we differentiate with respect to time: \[ \frac{dA}{dt} = \frac{d}{dt}(\pi r^2) \]
• Chain rule: \[ \frac{d}{dt}(r^2) = 2r \frac{dr}{dt} \]
• Hence: \[ \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \]

Step 1: Understanding the statement
The statement says: \[ \text{Area expands } 3c \text{ times as fast as radius} \] This means: \[ \frac{dA}{dt} = 3c \cdot \frac{dr}{dt} \]

Step 2: Differentiate area
\[ A = \pi r^2 \] Differentiating both sides w.r.t. time \(t\): \[ \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \]

Step 3: Equate given relation
From Step 1 and
Step 2: \[ 2\pi r \frac{dr}{dt} = 3c \frac{dr}{dt} \]

Step 4: Cancel common factor
Since \( \frac{dr}{dt} \neq 0 \), divide both sides: \[ 2\pi r = 3c \]

Step 5: Substitute given radius
Given: \[ r = 6 \] So: \[ 2\pi (6) = 3c \] \[ 12\pi = 3c \]

Step 6: Solve for \(c\)
\[ c = \frac{12\pi}{3} = 4\pi \]

Step 7: Final Answer
\[ \boxed{4\pi} \]