The area A of a circle is increasing at a constant rate of 1.5 cm ^2s^ -1 . Find, to 3 significant figures…

Analysis

We are given the rate of change of the area of a circle with respect to time ($\frac{dA}{dt} = 1.5 \text{ cm}^2\text{s}^{-1}$) and asked to find the rate of change of the radius ($\frac{dr}{dt}$) at the specific moment when the area $A = 2 \text{ cm}^2$.

Concept

The area of a circle is given by the formula:

To find the relationship between the rates of change, we differentiate both sides with respect to time ($t$) using the chain rule:

Step-by-Step Solution

Step 1: Find the radius $r$ when $A = 2 \text{ cm}^2$ Using $A = \pi r^2$:

Step 2: Substitute the known values into the differentiated formula We know $\frac{dA}{dt} = 1.5$ and $r \approx 0.79788$.

Step 3: Solve for $\frac{dr}{dt}$

Rounding to 3 significant figures, we get 0.299 cms$^{-1}$.

Conclusion

Comparing our result to the given options, the correct value is 0.299. This matches option D.

Exam Tip

When dealing with "related rates" problems in JAMB, always write down the geometric formula first (like $A = \pi r^2$ or $V = \frac{4}{3}\pi r^3$), differentiate it with respect to time $t$, and then plug in your specific values. Don't plug in the numbers before differentiating, or you will lose the variable you are trying to solve for!