related rates problem

Question

A conical water tank with vertex down has a radius of 10 feet at the top and is 28 feet high. If water flows into the tank at a rate of 30 , how fast is the depth of the water increasing when the water is 17 feet deep?

Answer 1

Sketch it out to get your head wrapped around the problem:

The rate at which the water flows in is the change in volume with respect to time, aka dV/dt

The volume at any point in time is given by:

V = 1/3πr²h

But that leaves us with an equation in 2 variables. In this case, we can use the geometry of what was given about the tank to write a relationship between height (h) and radius (r). Taking a cross section of the tank:

r=10

________

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|_r_w_ /

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The entire height is 28 and we can call the height of the water at any point h_w and since there are two similar triangles here, we can write a proportion:

28/h_w = 10/r_w

28r_w = 10h_w

r_w = 10h_w/28 = 5h_w/14

So, in our volume equation, we can replace "r" with "5h/14":

So V = 1/3π(5h/14)²h

V = 1/3π(25/196)h³

V = (25π/588)h³

Taking the derivative (using the chain rule):

dV/dt = (75π/588)h²•dh/dt

dV/dt = (25π/196)h²•dh/dt

Now, plug in the numbers, dV/dt = 30, h = 17 and solve for dh/dt