Question

Historically, there have been several major oil spills. Wild life suffers greatly and the ecosystem is thrown into disarray. Time is of ultimate concern and goals are to stop the flow of oil from the tanker and confine the oil spill using floating containment booms as quickly as possible. An oil tanker has gone aground on a reef and has punctured the oil tank. Oil is now flowing out of the tanker and creating a circular oil slick that is about 0.1 foot thick and the radius was calculated to be increasing at 0.32 foot/minute when the radius of the circle was at 500 feet. Note: a typical oil tank is about 330 yards x 60 yards x 30 yards and is close to a rectangular box.

A.) In general, at what rate is the volume of the ship’s oil decreasing? What is the rate of change of the volume when the spill is at 500 feet?

B.) If the oil were to leak at an average rate of that found in part A, how long will it take if the hole in the tank is not repaired before all of the oil has leaked out? (Give answer in number of days.) Note: the hole is in the very bottom of the tank.

C.) A little bit of good news. The hole was quite small and it has been plugged. The oil has stopped leaking. Now we must work fast to confine the oil that has leaked. How many feet of containment booms will be needed if we stopped the oil spill when the tank was still half full? Put into perspective how big the oil spill is – how far across the spill is it?

Answer 1

This is a calculus question.

A) Rate of volume of the ship's oil decreasing = rate of the oil flowing out increasing
= volume of circular slick increasing
dV/dt = dV/dr * dr/dt

dr/dt is given as 0.32foot/min when r = 500 ft. So just find dV/dr.

Volume, V = area * depth = pi*r^2 * 0.1
dV/dr = 2pi *r *0.1 = 0.2pi*r

Substitute r=500,
dV/dr = 0.2*pi*500 = 100pi

dV/dt= 100pi * 0.32 = 32 feet cubed/min

B) (note: 1 yard is 3 feet).

Calculate total volume of oil in the ship.
330 yards* 60 yards* 30 yards
=990 feet * 180 feet * 90 feet (convert all to feet because the answer in A is given in feet cubed)
=16038000 cubic feet.

Total number of min the oil leaked for, 16038000/32 = 501187.5 min.
(question require it to convert to number of days)
501187.5/60 = 8353.125 hours
8353.125/24 =348 days.

C) when the ship leak till its half empty. That means 16038000/2= 58019000 cubic feet of oil will be leaked out.
At a depth of 0.1 foot thick, 58019000/0.1 = 580190000 squared feet. This is the area of the circular area, the oil will cover.

Find the radius to have an idea of how big it actually is,
sqrt(580190000/ pi) = 13590 feet.

Answer 2

how is 100pi * .32 32 ft cubed per minute?

Answer 3

Opps i lacked the pi there. it should be 32pi ft cubed/min.

Argh that means the following are all wrong. Please make the appropriate changes.

Answer 4

No idea how big one containment boom is.