Question

If the radius of a circle increases at a uniform rate of \(2\ \text{cm/s}\), then the rate of increase of area of the circle, at the approximate instant when the radius is \(20\ \text{cm}\), is:

• A. \(80\pi\ \text{m}^2/\text{s}\)
• B. \(80\ \text{m}^2/\text{s}\)
• C. \(80\pi\ \text{cm}^2/\text{s}\)
• D. \(80\ \text{cm}^2/\text{s}\)

Hint:

In related rates problems: [ A=pi r^2 Rightarrow fracdAdt=2pi r fracdrdt ] Always substitute numerical values emphafter differentiation.

Answer 1

The Correct Option is C

Step 1: Write the formula for the area of a circle. \[ A = \pi r^2 \] Step 2: Differentiate both sides with respect to time \(t\). \[ \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \] Step 3: Substitute the given values. \[ \frac{dr}{dt} = 2\ \text{cm/s}, \quad r = 20\ \text{cm} \] \[ \frac{dA}{dt} = 2\pi \times 20 \times 2 \] Step 4: Simplify. \[ \frac{dA}{dt} = 80\pi\ \text{cm}^2/\text{s} \] Step 5: Final conclusion. The rate of increase of the area of the circle is: \[ \boxed{80\pi\ \text{cm}^2/\text{s}} \]