DIFF CALCULUS

A swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3 meters deep at the deep end. Water is being pumped into the pool at ¼ cubic meter per minute, and there is 1 meter of water at the deep end.
a. What percent of the pool is filled?
b. At what rate is the water level rising?

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  1. Part b is easy and requires no assumptions about the shape of the pool bottom.

    The rate of change of height = volume added per second / surface area of water.
    (draw a picture)
    or
    dh/dt = .25 m^3/min / (12m*6m)
    = .00347 m/min or 21 cm/hour

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  2. now the volume is another story because now we will have to make some assumption, for example constant slope up from 3 m to 1 m. In that case we have a trapezoid and
    V = 6 * 12 * average depth
    = 6 * 12 * 2 - 144 m^3

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