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9 May, 08:42

A conical tank has a radius of 2 ft at the top and a height of 3 ft. If a liquid flows in a rate of 2 ft^3/min, how fast is the water level rising when it is 2 ft high?

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  1. 9 May, 08:45
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    0.358 ft/min

    Step-by-step explanation:

    Let's say r and h are the radius and height of the water, and R and H are the radius and height of the tank.

    The volume of the water is:

    V = π/3 r² h

    Using similar triangles:

    r / h = R / H

    r = h R/H

    Substitute:

    V = π/3 (h R/H) ² h

    V = π/3 (R/H) ² h³

    Take derivative with respect to time:

    dV/dt = π (R/H) ² h² dh/dt

    Plug in values:

    2 = π (2/3) ² (2) ² dh/dt

    2 = 16π/9 dh/dt

    1 = 8π/9 dh/dt

    dh/dt = 9 / (8π)

    dh/dt ≈ 0.358

    The water is rising at approximately 0.358 ft/min.
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