Conical Tank Water Leak Calculus Problem

Question

Water is leaking out of an inverted conical tank at a rate of 14100.0 cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6.0 meters and the diameter at the top is 6.5  meters. If the water level is rising at a rate of 22.0 centimeters per minute when the height of the water is 3.0  meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute. __________



Answer 1

Let

h (function of time) be the height of water in the tank,

r (function of time) be the radius of the water surface in the tank,

k = 3.25/6 be the ratio r/h, (It is the same constant everywhere in the tank.)

V = πr²h/3 (function of time) be the volume of the water in the tank,

h', r', V' denote derivatives with respect to time of h, r, V respectively.

We are going to calculate the rate at which the amount of water in the tank keeps increasing,

i.e., V'.

Once we know V' then the rate of pumping equals V' increased by the rate of leaking.

Using the constancy of the ration r/h, we can write

r = kh

Substitute into the formula for V

V = πk²h³/3

Differentiate with respect to time

V' = πk²h²h'

Substitute h = 300 cm, h' = 22 cm/min

V' = π×(3.25/6)²×300²×22 = 1825069 cm³/min

Thus the water must be pumped at the rate

1825069 + 14100 = 1839169 cm³/min