Question

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Water is stored in a conical tank with a faucet at the bottom. The tank has depth 24 in. and radius 9 in., and it is filled to the brim. If the faucet is opened to allow the water to flow at a rate of 5 cubic inches per​ second, what will the depth of the water be after 4 ​minutes?

Answer 1

Depth of the water after 4 minutes = 18.15 inches

Explanation:

1. Convert 4 minutes to seconds: 4 minutes * 60 seconds/minute = 240 seconds.

2. Determine the volume of the tank: Since it is a conical tank, we can use the formula for the volume of a cone: V = (1/3) * π * r^2 * h, where V is the volume, π is a constant (approximately 3.14), r is the radius, and h is the height or depth of the cone.

V = (1/3) * 3.14 * (9^2) * 24 = 6,076.32

3. Calculate the rate of water flow:

Multiply the flow rate by the time: 5 cubic inches/second * 240 seconds = 1,200 cubic inches.

4. Subtract the amount of water that has flowed out from the initial volume of the tank: 6,076.32 cubic inches - 1,200 cubic inches = 4,876.32 cubic inches.

5. Determine the new depth of the water: To find the new depth, we need to rearrange the formula for the volume of a cone to solve for h:

V = (1/3) * π * r^2 * h

Rearranging, we get: h = 3V / (π * r^2)

Plugging in the values, we have: h = 3 * 4,876.32 / (3.14 * (9^2)) = 18.15 inches.