Question

1. calculator allowed the radius of a circle is decreasing at a constant rate of 0.1 centimeter per second. in terms of the circumference c, what is the rate of change of the area of the circle, in square centimeters per second? (a) (-(0.2)pi c) (b) (-(0.1)c) (c) (-\frac{(0.1)c}{2pi}) (d) ((0.1)^2 c) (e) ((0.1)^2 pi c)

Explanation:

Step1: Recall formulas for area and circumference of a circle

The area of a circle is \( A = \pi r^2 \), and the circumference is \( C = 2\pi r \). We need to find \( \frac{dA}{dt} \) in terms of \( C \), given \( \frac{dr}{dt} = - 0.1 \) cm/s (negative because radius is decreasing).

Step2: Differentiate the area with respect to time

Using the chain rule, \( \frac{dA}{dt} = \frac{d}{dt}(\pi r^2) = 2\pi r \frac{dr}{dt} \).

Step3: Substitute \( 2\pi r \) with \( C \)

From the circumference formula, \( C = 2\pi r \), so substitute \( 2\pi r \) in the derivative of area: \( \frac{dA}{dt} = C \cdot \frac{dr}{dt} \).

Step4: Plug in the value of \( \frac{dr}{dt} \)

We know \( \frac{dr}{dt} = - 0.1 \) cm/s, so \( \frac{dA}{dt} = C \cdot (-0.1) = - (0.1)C \).

Answer:

B. \(-(0.1)C\)