SOLUTION: Water is being poured at the rate of 15cm3/sec into an inverted cone with a diameter of 3m and a height of 6m. At what rate is the surface rising just as the tank is filled?

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Question 1193216: Water is being poured at the rate of 15cm3/sec into an inverted cone with a diameter of 3m and a height of 6m. At what rate is the surface rising just as the tank is filled?
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The volume of the cone is %281%2F3%29%28pi%29%28r%5E2%29h

We are given dV/dt; we need to determine dh/dt. So we need to express the volume as a function of h only.

The full cone has height 6 and diameter 3, so its radius is 1.5. Since the sides of the cone are straight, the radius is always 1/4 of the height.

r=h%2F4


15=dV%2Fdt=%28dV%2Fdh%29%28dh%2Fdt%29=%28%281%2F16%29%28pi%29h%5E2%29%28dh%2Fdt%29

dh%2Fdt=240%2F%28%28pi%29h%5E2%29

At the moment the cone is filled, the height h is 6, and

dh%2Fdt=240%2F%2836pi%29=20%2F%283pi%29

ANSWER: Just as the tank is filled, the water is rising at a rate of 20%2F%283pi%29 cm/sec