related rates calculus

Question

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

Answer 1

Let leakout = 8900 cc/min. Height of the cone = 7m radius of the cone at top = 2.75m Volume of the cone

V = πr²h/3. Using similar triangles in the cone r/2.75 =h/7 so r= 2.75h/7 now V= πh³(2.75/7)²/3 V=πh³(.154).

V= .051πh³ and dV/dt = .153πh²dh/dt where dh/dt = 18c/min. This now yields dV/dt = 175.2x10⁴cc/min for the fill. To find the rate at which water is being pumped into the tank add the leak rate so that the rate is equal

to 1.75x10⁴ cc/min + .89x10⁴cc/min which gives 176.09x10⁴ cc/min