Oil spills out of a tanker at the rate of 145 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 90 feet.

Question

Quizplus · Accepted Answer

The volume of oil spilled per minute is 145 gallons. Converting gallons to cubic feet using the conversion factor (1 cubic foot = 7.5 gallons), we get the volume of oil spilled per minute in cubic feet as $145 \div 7.5$. The oil forms a cylindrical shape where the volume $V$ can be expressed as $V = \pi r^2h$, where $r$ is the radius and $h$ is the height (or thickness) of the oil spill. Given the thickness $h = \frac{1}{4}$ inch, we convert this to feet ($\frac{1}{4} \div 12$ feet since there are 12 inches in a foot). To find the rate at which the radius of the spill is increasing, we differentiate the volume with respect to time, $dV/dt = 2\pi rh(dr/dt)$, and solve for $dr/dt$, the rate of change of the radius. Substituting the given values and solving for $dr/dt$ gives us the rate at which the radius is increasing when the radius is 90 feet.

Oil Spills Out of a Tanker at the Rate of 145