find the rate at which water is being pumped into the tank in cubic centimeters per minute.

Question

Water is leaking out of an inverted conical tank at a rate of 8200.0cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height 15 meters and the diameter at the top is 7 meters. If the water level is rising at a rate of 20.0 centimeters per minute when the height of the water is 4 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute.

Answer 1

Let R_out be the volumetric rate of flow out = 8200 cm³/min and R_iin , the unknown input volumetric rate.

We are told that h = 15 meters when diameter is 7 meters. It is a cone-shaped tank, so we know that this ratio is constant h/d = 15/7. If dh/dt = 20 cm/min when h = 400 cm (make sure all length units mesh with volumes), what is R_in?

From the accumulation of volume equation dV/dt = R_in - R_out and R_in = dV/dt + R_out

From the volume equation for a cone: 1/3 πr²h = V, 1/12 πd²h = V, 1/12 * (7/15)²h³ = V using r = d/2 and d = 7/15 h.

Related rates --> take the derivative of the V equation to relate dV/dt to dh/dt:

(3/12)* (7/15)²h²dh/dt = dV/dt (You know everything on the left-hand side and you can use the accumulation equation to solve for R_in