The radius of a circle is increasing at the rate of 0.02cms-1. Find the rate at which the area is increasing when the radius of the circle is 7cm.

Analysis

This is a problem involving Related Rates in Calculus. We are given the rate of change of the radius with respect to time ($\frac{dr}{dt}$) and asked to find the rate of change of the area with respect to time ($\frac{dA}{dt}$) at a specific radius ($r = 7\text{cm}$).

Concept

1. Formula for the area of a circle: $A = \pi r^2$
2. Chain Rule: Since $A$ is a function of $r$, and $r$ is a function of $t$, we differentiate $A$ with respect to $t$: $$\frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt}$$

Step-by-Step Solution

Step 1: Identify the given values.

• Rate of change of radius: $\frac{dr}{dt} = 0.02 \, \text{cm/s}$
• Radius at the specific moment: $r = 7 \, \text{cm}$
• Take $\pi \approx \frac{22}{7}$ (standard for JAMB calculations involving multiples of 7).

Step 2: Differentiate the area formula. Given $A = \pi r^2$, differentiate with respect to $r$:

Step 3: Apply the Chain Rule.

Step 4: Substitute the values.

The $7$ in the numerator and denominator cancels out:

Conclusion

Comparing our result to the options provided, the calculated rate of increase is $0.88 \, \text{cm}^2\text{s}^{-1}$. This matches option D.

Exam Tip

Whenever you see "rate of change" in a JAMB question, think Calculus. If you are given a geometric shape, write down the formula for its area or volume first, then differentiate it with respect to time ($t$) using the chain rule. Always look for ways to cancel out $\pi$ or fractions like $7$ to make your mental math faster!