Oil spills out of a tanker at the rate of 125 gallons per minute. The oil spreads in a circle with a thickness of 1/4 inch. Given that 1 ft³ equals 7.5 gallons, determine the rate at which the radius of the spill is increasing when the radius reaches 50 feet.

Question

Quizplus · Accepted Answer

The volume of oil spilled is related to the radius of the circular spill by the formula for the volume of a cylinder, $V = \pi r^2h$, where $r$ is the radius and $h$ is the height (or thickness) of the oil layer. Given the thickness $h = \frac{1}{4}$ inch = $\frac{1}{48}$ feet (since there are 12 inches in a foot), and the rate of oil spillage is 125 gallons per minute, which needs to be converted to cubic feet per minute using the conversion factor $1 \text{ gallon} = \frac{1}{7.5} \text{ cubic feet}$, we get the rate of volume increase as $125 \times \frac{1}{7.5}$ cubic feet per minute.Differentiating both sides of the volume formula with respect to time $t$ gives us $\frac{dV}{dt} = 2\pi r \frac{dr}{dt}h$. Substituting the given and derived values into this equation allows us to solve for $\frac{dr}{dt}$, the rate at which the radius of the spill is increasing when the radius is 50 feet.

Oil Spills Out of a Tanker at the Rate of 125