# the area of a circular oil slick on the surface of the sea is increasing at a rate of `150m^2/s` how fast is the radius changing when: a) the radius is 25 m b) the area is `1000 m^2`

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tjbrewer | Certified Educator

The conditions of a) and b) are a trick question, because the area is changing at a constant rate.

The oil slick is circular so it's area is calculated by the formula `A=pi r^2` Pi is a constant, and we know that A is increasing at a constant rate of `150m^2/s` So the rate of change in r is a constant that would result in `A=150`

We enter the rate of change as Area to find the value of r. `150=pi r^2` We isolate the variable r and we get `r^2=150/pi` We solve for r and we get `r=sqrt(150/pi)` The radius of the oil slick is growing at the rate of `sqrt(150/pi) m/s~~6.91m/s`

I hope you were able to follow all that. The answer to each subquestion is the same. Roughly 6.91 m/s `sqrt(150/pi)` to be exact.