Patrick D. answered • 04/12/17

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Patrick the Math Doctor

No data is given about the rate that the radius is changing as a function of time (that is dR/dt).

Examining the 2-D version of the inverted cone, slicing it in half along the height, the height is 1200 cm when the radius is 250 cm as given in the problem. So for any given height h, the radius at that height forms two similar right triangles.

The proportion is h/1200 = r/250 --> r = 5*H/24

Substituting this into the formula for the volume of the cone:

V = 1/3*PI* R^2*H = 1/3*Pi* (5/24H)^2 * H = 1/3*pi* 25/576*H^3

Taking the derivative of both sides:

dV/dt = pi * 25/576*H^2* dH/dt

As given in the problem, dV/dt = c - 7100 for some fixed constant we are trying to find.

H = 4.5m = 450 cm when dH/dt = 28 cubic centimeters per min

The equation becomes:

c-7100 = pi * 25/576 * 450 ^2 * 28

Solving for c and using pi=22/7 gives a result of 780537.5 cubic centimeters per minute or .7805575 meters per minute.

At this rate the tank will fill in just over minutes