A ruptured oil tanker causes a circular oil slick on the surface of the ocean. W

Question

A ruptured oil tanker causes a circular oil slick on the surface of the ocean. When its radius is 150 meters, the radius of the slick is expanding by 0.1 meter/minute and its thickness is 0.02 meter. At that moment: How fast is the area of the slick expanding? The circular slick has the same thickness everywhere, and the volume of oil spilled remains fixed. How fast is the thickness of the slick decreasing? A property owner wants to fence a garden adjacent to a road. The fencing next to the road must be sturdier and costs $5 per foot, but the other fencing costs just $3 per foot. The garden is to have an area of 1200 square feet. Find the function that models the cost of fencing the garden. Find the garden dimensions that minimize the cost of fencing. If the owner has at most $600 to spend on fencing, find the range of lengths he can fence along the road.

Explanation / Answer

When r = 150 m and h = 0.02 m, dr/dt = 0.1 m/min

(a) A = r^2

dA/dt = (2 r)(dr/dt) = 2 * 150 * 0.1 = 94.25 m^2 /min

(b) V = r^2 h

dV/dr = [r^2 (dh/dt) + h * 2r (dr/dt)]

0 = [150^2 (dh/dt) + 2 * 0.02 * 150 (0.1)]

dh/dt = -2 * 0.02 * 150 * 0.1 / 150^2 = -0.000027 m/min

The thickness is decreasing at the rate of 0.000027 m/min.

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