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An oil tanker traveling in the ocean is leaking and causes a circular oil slick on the surface of the water. The oil leaks at a rate of 4 gallons per minute. Assume the thickness of the leak is uniformly 0.5 inches.

At what rate is the radius of the leak increasing when 80 gallons of oil have spilled out?

Note: 1 gallon = 231 cubic inches.

Answer :

The rate at which the radius of the oil slick is increasing when 80 gallons of oil have spilled out is dr/dt = (4 * 231 * 80) / (π * 0.5 * r_initial).

For solving this problem, we can use the concept of related rates from calculus. The volume of the oil slick can be approximated as a cylinder with a circular base, and we want to find the rate at which the radius of the circular oil slick is increasing when 80 gallons of oil have spilled out.

The information:

Oil leak rate: 4 gallons/minute

Thickness of the leak: 0.5 inch

Volume of 1 gallon: 231 cubic inches

Desired volume of spilled oil: 80 gallons * 231 cubic inches/gallon

Let's denote the radius of the circular oil slick as "r" (in inches) and the height (thickness) of the oil slick as "h" (in inches).

The volume of a cylinder is given by the formula:

Volume = π * r^2 * h

Since the thickness of the oil slick is uniform and equal to 0.5 inches, we have:

h = 0.5 inches

The volume of the spilled oil is:

Volume = π * r^2 * 0.5 inches

We're given that the oil leaks at a rate of 4 gallons/minute, which can be converted to cubic inches/minute:

Leak rate = 4 gallons/minute * 231 cubic inches/gallon

We want to find the rate at which the radius (r) is increasing when 80 gallons of oil have spilled out, i.e., when the volume is equal to 80 gallons * 231 cubic inches/gallon.

Let's differentiate the volume equation with respect to time (t), using the chain rule:

d(Volume)/dt = π * 2r * dr/dt * h

Now we can plug in the given values:

d(Volume)/dt = 4 gallons/minute * 231 cubic inches/gallon

Volume = π * r^2 * 0.5 inches

Desired volume = 80 gallons * 231 cubic inches/gallon

We can solve for dr/dt, the rate at which the radius is increasing, when the volume is the desired volume.

Solving for dr/dt:

π * 2r * dr/dt * 0.5 = 4 * 231 * 80

r * dr/dt = (4 * 231 * 80) / (π * 0.5)

dr/dt = (4 * 231 * 80) / (π * 0.5 * r)

Now, substitute the value of "r" when the desired volume is reached:

dr/dt = (4 * 231 * 80) / (π * 0.5 * r_initial)

This will give you the rate at which the radius of the oil slick is increasing when 80 gallons of oil have spilled out. Make sure to convert the final answer to the appropriate units based on your initial unit of "r" (in inches).


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