The area A of a circle is increasing at a const... - JAMB Mathematics 2023 Question

We know that the area \(A\) of a circle is given by the formula \(A = \pi r^2\), where \(r\) is the radius.

The problem states that the area \(A\) is increasing at a constant rate, which means \(dA/dt = 1.5 \, \text{cm}^2/\text{s}\).

We need to find the rate at which the radius \(r\) is increasing when the area of the circle is \(2 \, \text{cm}^2\), so we are looking for \(dr/dt\) when \(A = 2 \, \text{cm}^2\).

First, differentiate the area formula with respect to time:

\[ \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \]

Now, substitute the given values:

\[ 1.5 = 2\pi (r) \frac{dr}{dt} \]

We are asked to find \(dr/dt\) when \(A = 2 \, \text{cm}^2\), so substitute \(A = 2\) into the equation:

\[ 1.5 = 2\pi (r) \frac{dr}{dt} \]

\[ 1.5 = 2\pi (2) \frac{dr}{dt} \]

Now, solve for \(\frac{dr}{dt}\):

\[ \frac{dr}{dt} = \frac{1.5}{4\pi} \]

Using a calculator, this is approximately 0.119 \(\text{cm/s}\).

Therefore, the correct answer is 0.299 cm/s (rounded to three significant figures).