[Math] Related Rate Question: Water is leaking out of an inverted conical tank at a rate of 9,500 cm3/min

calculus

Water is leaking out of an inverted conical tank at a rate of 9,500 cm3/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, find the rate at which water is being pumped into the tank. (Round your answer to the nearest integer)

My answer was 288,752 cm3/min

However, the answer is wrong. Can someone help me?

Best Answer

Let $h$ denote the height, and $r$ the radius of the part of the cone that is filled with water. Since the total height is 6 and the radius at the top is 2, using proportions we have $r=\frac h3$. The volume of cone is $V=\pi r^2 h/3 = \pi h^3/27$. Here both $V$ and $h$ are functions of the time $t$, so differentiating both sides with respect to $t$ we get $V'=3h^2\pi h'/27=h^2\pi h'/9$. It is given that $h'=20$ when $h=2 m = 200 cm$, so we have $V'= 200^2\cdot 20\pi /9 \approx 279252.68$. Adding the 9500 we get $288752.68\approx 288753$. (It appears to me you got the answer mostly right, except the "Round your answer to the nearest integer" part.)