Thread: Related Rate

The bad news: a tanker accident has spilled oil in Pristine Bay. The good news: oil eating- bacteria are gobbling up 5 cubic feet per hour. The slick takes the form of a cylinder, whose height is the thickness of the slick. When the radius takes the form of a cylinder, whose height is the thickness of the slick. When the radius of the cylinder is 500 feet, the thickness of the slick is .o01ft, and decreasing at .001 ft/hr. At this time, what is the rate at which the area of the slick increasing (could be a typo and is meant to be decreasing?)

Originally Posted by BarlowBarlow1

The bad news: a tanker accident has spilled oil in Pristine Bay. The good news: oil eating- bacteria are gobbling up 5 cubic feet per hour. The slick takes the form of a cylinder, whose height is the thickness of the slick. When the radius takes the form of a cylinder, whose height is the thickness of the slick. When the radius of the cylinder is 500 feet, the thickness of the slick is .o01ft, and decreasing at .001 ft/hr. At this time, what is the rate at which the area of the slick increasing (could be a typo and is meant to be decreasing?)

ALWAYS DRAW A DIAGRAM FOR RELATED RATES PROBLEMS!

See the diagram below:

Let V be the volume of the cylinder
Let h be the height of the cyliner
Let r be the radius of the cylinder
Let A be the area of the cylinder

V = A*h
=> A = V/h
differentiating implicitly by the Quotient rule, we get:

dA/dt = (h*dV/dt - V*dh/dt)/h^2

now we know what h is, we know what dV/dt is, we know what dh/dt is...but what is V at this instant?


Now, for a cylinder:

V = (area of base)*height
=> V = pi*r^2*h
at the instant we are concerned with, r = 500 and h = 0.001
=> V = pi*(500)^2 *0.001 = 250pi

now we can find dA/dt

dA/dt = (h*dV/dt - V*dh/dt)/h^2
.........= [0.001*(-5) - (250pi)*(-0.001)]/(0.001)^2
.........= (-0.005 + 0.25pi)/(0.000001)
.........~= 7.8 x 10^5 ft^2/hour

so from this, it would seem the area is increasing...which makes o sense to me either check my computation

Originally Posted by BarlowBarlow1

The bad news: a tanker accident has spilled oil in Pristine Bay. The good news: oil eating- bacteria are gobbling up 5 cubic feet per hour. The slick takes the form of a cylinder, whose height is the thickness of the slick. When the radius takes the form of a cylinder, whose height is the thickness of the slick. When the radius of the cylinder is 500 feet, the thickness of the slick is .o01ft, and decreasing at .001 ft/hr. At this time, what is the rate at which the area of the slick increasing (could be a typo and is meant to be decreasing?)

The reason that the area can be increasing while the volume is decreasing
is because the slick is spreading. So even without the bacteria the slick
can thin and spread. In this problem the area is not changing only because
the bacteia are digesting the oil.

RonL