The area of a circular sun spot is growing at a rate of 1,200 km²/s. How fast is the radius growing at the instant when it equals 10,000 km? A)8.333 km/s B)0.000 km/s C)0.038 km/s D)0.019 km/s E)0.060 km/s

Question

Quizplus · Accepted Answer

We have the formula for the area of a circle: $A=\pi r^2$. Differentiating both sides with respect to time $t$, we get:
$$\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$$
We are given that $\frac{dA}{dt} = 1200 	ext{ km}^2/	ext{s}$. When the radius is 10,000 km, we want to find $\frac{dr}{dt}$. Plugging in these values into the formula above, we get:
$$1200 = 2\pi (10000) \frac{dr}{dt}$$
Solving for $\frac{dr}{dt}$, we get:
$$\frac{dr}{dt} = \frac{1200}{2\pi (10000)} \approx 0.019 	ext{ km/s}$$
Therefore, the answer is (D).

The Area of a Circular Sun Spot Is Growing at a Rate