Question

A circular plate of radius 5 cm is heated. Due to expansion, its radius increases at the rate of 0.05 cm/sec. The rate at which its area is increasing when the radius is 5.2 cm is

• A. 5.05 π cm²/sec
• B. 5.2 π cm²/sec
• C. 0.52 π cm²/sec
• D. 27.4 π cm²/sec

Hint:

To find the rate at which the area of a circle is increasing, differentiate the area formula \( A = \pi r^2 \) with respect to time. Use the chain rule to account for the changing radius. The result involves multiplying the radius, the rate of change of the radius, and \( \pi \). Remember to substitute the given values for the radius and the rate of change of the radius at the specific point in time.

Answer 1

The Correct Option is C

The correct answer is (C) : 0.52 π cm²/sec.

Answer 2

The correct answer is: (C) 0.52 \( \pi \) cm²/sec.

We are given a circular plate with radius \( r = 5 \) cm. The radius is increasing at a rate of \( \frac{dr}{dt} = 0.05 \) cm/sec. We are asked to find the rate at which the area of the plate is increasing when the radius is \( r = 5.2 \) cm.

Step 1: Write the formula for the area of a circle

The area \( A \) of a circle is given by the formula: \[ A = \pi r^2 \] Step 2: Differentiate with respect to time

To find the rate at which the area is increasing, we differentiate \( A = \pi r^2 \) with respect to time \( t \): \[ \frac{dA}{dt} = 2 \pi r \frac{dr}{dt} \] Step 3: Substitute the known values

We are given that \( \frac{dr}{dt} = 0.05 \) cm/sec, and we want to find \( \frac{dA}{dt} \) when the radius \( r = 5.2 \) cm. Substituting these values into the derivative: \[ \frac{dA}{dt} = 2 \pi (5.2) (0.05) \] Step 4: Simplify the expression

\[ \frac{dA}{dt} = 0.52 \pi \, \text{cm}^2/\text{sec} \] Therefore, the rate at which the area is increasing is \( 0.52 \pi \) cm²/sec.

Thus, the correct answer is (C) 0.52 \( \pi \) cm²/sec.