math - calc

A conical water tank with vertex down has a radius of 12 feet at the top and is 26 feet high. If water flows into the tank at a rate of 30 {\rm ft}^3{\rm /min}, how fast is the depth of the water increasing when the water is 12 feet deep?

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  1. Make your sketch,
    let the radius of the water level be r ft
    let the height of the water be h ft

    by ratios:
    r/h = 12/26
    26r = 12h
    r = 6h/13

    Vol = (1/3)π r^2 h
    = (1/3)π (36h^2/169) g
    = (12/169)π h^3
    d(Vol)/dt = (36/169)π h^2 dh/dt

    30 = (36/169)π (144) dh/dt
    dh/dt = 30(169/(36π(144))
    = 845/(864π) ft/min = appr .3113 ft/min

    check my arithmetic, I should have written it down instead of doing on the screen only

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