Question

The area of a circular sun spot is growing at a rate of 1,400 km^2/s.
(a) How fast is the radius growing at the instant when it equals 8,000 km?
(b) How fast is the radius growing at the instant when the sun spot has an area of 640,000 km^2?

Answer 1

So we know the following:\[\bf A=\pi r^2\]\[\bf \frac{dA}{dt}=1,400\ km^2/s\]To find dr/dt, we must differentiate both sides of the Area equation:\[\bf \frac{ dA }{ dt }=2 \pi r \frac{ dr }{ dt }\]To answer a), we plug in the already given value of dA/dt and the the value of 8,000 km for the radius to get:\[\bf 1400 \ km^2/s=2 \pi (8000 \ km)\frac{ dr }{ dt } \implies \frac{ dr }{ dt } \approx 0.028 \ km/s\]To solve b.), we do essentially the same thing as a.) except this time we are not given the radius, instead we are given the Area. So we will plug in the given Area value, solve for the radius, and then plug that into the dA/dt equation and solve for dr/dt:\[\bf 640,000 \ km^2=\pi r^2 \implies r \approx 451.352 \ km\]Plug this in to dA/dt and solve for dr/dt:\[\bf 1,400 \ km^2/s=2 \pi (451.352 \ km)\frac{ dr }{ dt } \implies \frac{ dr }{ dt }\approx 0.494 \ km/s\]And we are done.

@lexusbreon

Answer 2

Thank you so much!!

Answer 3

Did I fan you now?

Answer 4

ya ty

Answer 5

no problem. Thanks again