Question
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A conical tank (with vertex down) is 28 ft across the top and 7 ft ft
deep. If water is flowing into the tank at the rate of , find
the rate of change of the depth of the water at the instant when it is 2
deep.

Ask by Boone Munoz. in Puerto Rico
Nov 01,2024

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Tutor-Verified Answer

Answer

The rate of change of the depth of the water when it is 2 ft deep is .

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Bonus Knowledge

To solve the problem, we first need to determine the relationship between the volume of water in the tank and its height. The tank forms a cone, and the volume of a cone is given by the formula .
Given that the tank is 28 ft across the top, the radius at the top is ft, and it is ft deep. Using the depths and corresponding radius, we find the relationship between the radius of the water’s surface and the height of the water.
Using similar triangles, we get:
Substituting into the volume formula gives:
Now, we need to differentiate this volume with respect to time :
We know that water is flowing into the tank at a rate of , and we want to find when .
Substituting :
Evaluating this gives us the rate of change of the depth of the water at that moment:
So, the rate of change of the depth of the water when it is 2 ft deep is approximately .

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