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A trough is 14 ft long and its ends have the shape of isosceles triangles that are 5 ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of 15 ft3/min, how fast is the water level rising when the water is 4 inches deep?

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MathPhys

Step-by-step explanation:

Using similar triangles, the height and width of the water is proportional to the height and width of the trough.

w / h = 5 / 1

w = 5h

The volume of the water is:

V = AL

V = (½ wh) (14)

V = 7wh

Substituting:

V = 7(5h)h

V = 35h²

Take derivative with respect to time:

dV/dt = 70h dh/dt

Given that dV/dt = 15 and h = ⅓:

15 = 70 (⅓) dh/dt

dh/dt = 9/14

dh/dt ≈ 0.643 ft/min

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