A conical water tank is 24 ft high and has a radius of 10 ft at the top. If water flows into the tank at a rate of 20 cubic feet per minute, how fast is the depth of the water increasing when the water is 16 feet deep?

Question

Advanced Placement (AP)

Gaige Newton

23 November, 17:16

A conical water tank is 24 ft high and has a radius of 10 ft at the top. If water flows into the tank at a rate of 20 cubic feet per minute, how fast is the depth of the water increasing when the water is 16 feet deep?

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Answers (1)

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Harper Medina · Answer 1

0.027 cubic ft/min

Explanation:

Volume of water is V=V (t) Depth of water is h=h (t)

The relationship between V and h is

V=π/3. r2. h

Let

h0 = the height of the cylinder = 24ft, r0 = the radius of the opening = 10ft.

Since water level (h) rise with respect to flow rate we find an expression for r and substitute into V

r/h = r0/h0 = 10/24 = 5/12

r = (5/12) h,

Substitute for r in V

so,

V=1/3.π. (h2) square = 1/3.π. 25/144. (h3)

When

h = 16ft

and dV/dt = 20ft3/min

,

dV/dt = 1/3*π*25/144*h3 * (dh/dt)

20ft3/min = 1/3*π * 25/144*h3 * (dh/dt)

20 = 1/3*π * (16) 3. (dh/dt)

Make dh/DT subject of formula

dh/dt = (20*3*144) / (25*π*4096) = 8640/321740.8

=

dh/dt = 0.027 ft3/min