Question 3 of 82intermediate🔧 ApplyNumerical3 marks

The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

Correct Answer

The area of the circle is increasing at the rate of 60π cm2/s60\pi \text{ cm}^2/\text{s}.

Exercise: EXERCISE 6.1 | Q: 3 | (Page 4)
For More Understanding

Explanation

This problem is solved using the application of derivatives as shown in Example 3 of the provided context. The area of a circle depends on the radius, and since the radius changes with time, the rate of change of area with respect to time is found using the Chain Rule. We differentiate the area formula A=πr2A = \pi r^2 with respect to tt to get dAdt=2πrdrdt\frac{dA}{dt} = 2\pi r \frac{dr}{dt}. Substituting the specific values given in the question (r=10r=10 and drdt=3\frac{dr}{dt}=3) yields the final answer.

Solution Steps

  1. Step 1: Let AA be the area of the circle and rr be the radius. The area is given by A=πr2A = \pi r^2.

  2. Step 2: Differentiate the area with respect to time tt using the Chain Rule: dAdt=ddt(πr2)=ddr(πr2)⋅drdt=2πrdrdt\frac{dA}{dt} = \frac{d}{dt}(\pi r^2) = \frac{d}{dr}(\pi r^2) \cdot \frac{dr}{dt} = 2\pi r \frac{dr}{dt}.

  3. Step 3: Substitute the given values into the equation. The radius is increasing at the rate of drdt=3 cm/s\frac{dr}{dt} = 3 \text{ cm/s}, and we need to find the rate when the radius is r=10 cmr = 10 \text{ cm}.

  4. Step 4: Calculate the rate of increase of area: dAdt=2π(10)(3)=60π\frac{dA}{dt} = 2\pi (10) (3) = 60\pi.

  5. Step 5: Conclude that the area is increasing at the rate of 60π cm2/s60\pi \text{ cm}^2/\text{s}.

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