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23 April, 20:09

g A circular oil slick of uniform thickness is caused by a spill of one cubic meter of oil. The thickness of the oil slick is decreasing at a rate of 0.1 centimeters per hour. At what rate is the radius of the slick increasing when the radius is 8 meters?

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  1. 23 April, 20:59
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    the rate of increase of radius is dR/dt = 0.804 m/hour = 80.4 cm/hour

    Explanation:

    the slick of oil can be modelled as a cylinder of radius R and thickness h, therefore the volume V is

    V = πR² * h

    thus

    h = V / (πR²)

    Considering that the volume of the slick remains constant, the rate of change of radius will be

    dh/dt = V d[1 / (πR²) ]/dt

    dh/dt = (V/π) (-2) / R³ * dR/dt

    therefore

    dR/dt = (-dh/dt) * (R³/2) * (π/V)

    where dR/dt = rate of increase of the radius, (-dh/dt) = rate of decrease of thickness

    when the radius is R=8 m, dR/dt is

    dR/dt = (-dh/dt) * (R³/2) * (π/V) = 0.1 cm/hour * (8m) ³/2 * π/1m³ * (1m/100 cm) = 0.804 m/hour = 80.4 cm/hour
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