Question

The rate of change of the area of a circular disc with respect to its circumference when radius is 3 is:

• A. 6 unit
• B. 3 unit
• C. 6π unit
• D. 2π unit

Answer 1

The Correct Option is B

To find the rate of change of the area of a circular disc with respect to its circumference when the radius is 3, we start by defining the following:

Let \( A \) be the area of the circle and \( C \) be the circumference. The area of a circle is given by:

\( A = \pi r^2 \)

The circumference is given by:

\( C = 2\pi r \)

We need to find \( \frac{dA}{dC} \), the rate of change of the area with respect to the circumference.

First, differentiate the area with respect to the radius:

\( \frac{dA}{dr} = 2\pi r \)

Next, differentiate the circumference with respect to the radius:

\( \frac{dC}{dr} = 2\pi \)

Apply the chain rule to find \( \frac{dA}{dC} \):

\( \frac{dA}{dC} = \frac{dA}{dr} \div \frac{dC}{dr} = \frac{2\pi r}{2\pi} = r \)

When \( r = 3 \),

\( \frac{dA}{dC} = 3 \)

Thus, the rate of change of the area with respect to its circumference when the radius is 3 is 3 unit.