SOLUTION: Water is pouring into an inverted cone at the rate of 8 cubic feet per minute. If the height of the cone is 12 ft and the radius of its base is 6 ft, how fast is the water level ri

Question 1197602: Water is pouring into an inverted cone at the rate of 8 cubic feet per minute. If the height of the cone is 12 ft and the radius of its base is 6 ft, how fast is the water level rising when the water is 4 ft deep?
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The volume is increasing at a rate of 8 cubic feet per minute; we want to know how fast the water level (height) is changing when the height is 8 feet.

Since we are given the rate of change of the volume and we want to find the rate of change of the height, we need a formula for the volume in terms of the height.

Volume of a cone:

The height of the cone is 12 feet and the radius of its base is 6 feet. Since the sides of the cone are straight, the ratio 6/12 = 1/2 of radius to height is constant. We want our formula in terms of height, so

And the volume of the cone in terms of the height is

Now we are ready to use our calculus for this related rates problem.

We want the rate of change of the height, , when the height is 4, knowing that the rate of change in the volume, , is 8:

ANSWER: The water level is rising at a rate of (2/pi) feet per second when the water is 4 feet deep.

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