Calculus Related Rates Question

Question

Water is leaking out of an inverted conical tank at a rate of 8500 cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height of 12  meters and the diameter at the top is 6.5  meters. If the water level is rising at a rate of  20 centimeters per minute when the height of the water is 4.5  meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute. 

Answer 1

Hi Andy,

We're trying to figure out the rate at which volume is being added to a tank. Translated to calculus terms, that means we're looking for dV/dt. Since the tank has the shape of a cone, it'd help to start off with an equation for the volume of a cone.

V = π/3*r²*h

We want dV/dt, which means we'll have to do implicit differentiation on both sides. We have two variables right now, which will make for an overly complicated derivative. We should either convert r in terms of h, or convert h in terms of r. Since the problem states "the water level is rising at 20 cm per minute", which is a fancy way of saying "dh/dt = 20", we should convert r to h.

To do this, you can use similar triangles. The tank has a height of 12 meters and the diameter at the top is 6.5 meters. That means the radius is 3.25 meters. So,

r / h = 3.25 / 12

r = 3.25*h / 12

r = 13h/ 48

We can now express the volume as strictly a function of h

V = π/3*(13h/48)²*h

V = 169π/6912 * h³

What a gross number.

Anyway, we can implicitly differentiate both sides of the equation with respect to t.

d/dt( V = 169π/6912 * h³ )

dV/dt = 169π/2304 * h² * dh/dt

Let's translate what these all mean.

dV/dt = rate of change of the volume. This is what we're looking for.

h = present height of the water level. h = 4.5m = 450cm

dh/dt = rate of change of the water level. dh/dt = 20 cm/min

Plug all of this in:

dV/dt = 169π/2304 * (450)² * 20

dV/dt = 933,274 cm³/ min

This is the overall rate of change of the volume. If you're leaking 8500 cm³/ min, then your pump must be going at:

933,274 cm³/ min + 8500 cm³/ min = 941,774 cm³/ min

These are unnecessarily complicated numbers. But I hope this helped.

Answer 2

The radius at the top is 3.25 m (1/2 the diameter), and because of similar triangles, the ratio of the radius to the height at any point on the cone stays constant. Thus h/r = 12/3.5 = 24/7 and r = 7h/24.

V_cone = 1/3πr²h and plugging in the expression above for r gives V as a function of h:

V(h) = 1/3π(49h²/576)h = 49πh³/1728

dV/dt = 49πh²/576·dh/dt Plugging in h = 4.5 and dh/dt = 20 gives

dV/dt ≈ 108 = Rate_in - 8,500

R_in ~ 8,608 m³/min