Question

Consider a circular oil slick that forms on the surface of a large lake. when the radius of the oil slick is 160 meters, the radius is expanding by 0.4 meter/minute. at that moment, how fast is the area of the slick expanding? convert your answer to square meters per second.

Answer 1

The area of the oil slick is expanding at a rate of 16.775 square meters per second.

Step-by-step explanation:

To find how fast the area of the slick is expanding, we can use the formula for the area of a circle, A = πr^2, where A is the area and r is the radius. Taking the derivative of both sides of the equation with respect to time, we get dA/dt = 2πr(dr/dt). Plugging in the known values r = 160 m and dr/dt = 0.4 m/min, we can calculate dA/dt as follows:

dA/dt = 2π(160)(0.4) = 320π = 1006.5 m^2/min

Since the question asks for the answer to be in square meters per second, we need to convert from minutes to seconds. Since there are 60 seconds in a minute, we can divide the answer by 60:

dA/dt = 1006.5/60 = 16.775 m^2/s