Calculus

A rectangular swimming pool is 8 m wide and 20 m long. Its bottom is a sloping plane, the depth increasing from 1 m at the shallow end to 3 m at the deep end. Water is draining out of the pool at a rate of 1 m3/min. How fast is the surface of the water falling when the depth of the water at the deep end is (a) 2.5 m? (b) 1 m?

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  1. When the water is 2.5m deep, the section draining is just a rectangular prism with area 160m^2 and height 1/2 m.

    So, draining at 1m^3/min, the water level is dropping at a rate of 1/120 m/min

    When the water is y m deep (y<2), the side cross-section of the pool is a right triangle with legs y and 10y.

    So, the volume of the water is

    v = (1/2)(y)(10y)(8) = 40y^2 m^3
    dv/dt = 80y dy/dt
    so, at y=1,
    -1 = 80*1 dy/dt
    dy/dt = -1/80 ft/min

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  2. typo alert
    dy/dt at y=2.5 is -1/160, not -1/120

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