SOLUTION: The diagram shows a vertical cross -section of a container in the form of invented cone of height 60cm and base radius 20cm. The circular base is held horizontal and uppermost. W

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Question 1187852: The diagram shows a vertical cross -section of a container in the form of invented cone of height 60cm and base radius 20cm. The circular base is held horizontal and uppermost. Water is pured into the container at a constant rate of 40cm^3/s.
i)show that, when the depth of water in the container is x cm, the volume in the container is (pie x^3)/27 cm^3
ii) find the rate of increase of x at the instant when x=2.
Found 3 solutions by Edwin McCravy, mccravyedwin, AnlytcPhil:
Answer by Edwin McCravy(20056) About Me  (Show Source):
Answer by mccravyedwin(407) About Me  (Show Source):
Answer by AnlytcPhil(1806) About Me  (Show Source):
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Because of similar right triangles,

r%2Fx%22%22=%22%2220%2F60
r%2Fx%22%22=%22%221%2F3
3r%22%22-%22%22x
r%22%22=%22%22expr%281%2F3%29x

The formula for the volume of a cone is 

V%22%22=%22%22expr%281%2F3%29%28pi%2Ar%5E2h%29

Substituting h=x and r=1%2F3,

V%22%22=%22%22expr%281%2F3%29%28pi%2A%28expr%281%2F3%29x%29%5E2%28x%29%29
V%22%22=%22%22expr%281%2F3%29%28pi%2Aexpr%281%2F9%29x%5E3%29
V%22%22=%22%22pi%2Ax%5E3%2F27%5E%22%22%29cm%5E3
ii) find the rate of increase of x at the instant when x=2.


dV%2Fdt%22%22=%22%22expr%28pi%2F27%293x%5E2%2Aexpr%28dx%2Fdt%29
dV%2Fdt%22%22=%22%22%28%28pi%2Ax%5E2%29%2F9%5E%22%22%29expr%28dx%2Fdt%29
40%22%22=%22%22%28%28pi%2A2%5E2%29%2F9%5E%22%22%29expr%28dx%2Fdt%29
40%22%22=%22%22%284pi%2F9%29expr%28dx%2Fdt%29

Divide both sides by 4

10%22%22=%22%22%28pi%2F9%29expr%28dx%2Fdt%29

Multiply both sides by 9%2Fpi

90%2Fpi%22%22=%22%22dx%2Fdt

Answer: The water level is rising at the rate of 90/π or about 28.6 centimeters per second.

Edwin