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Math Help - rates of change

  1. #1
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    rates of change

    Sand falls from a hopper at a rate of 0.6 cubic meters per hour and forms a conical pile beneath. Suppose the radius of the cone is always half the height of the cone.

    (a) Find the rate at which the radius of the cone increases when the radius is 2 meters.
    dr/dt = meters per hour

    (b) Find the rate at which the height of the cone increases when the radius is 2 meters.
    dh/dt = meters per hour


    and


    A ruptured oil tanker causes a circular oil slick on the surface of the ocean. When its radius is 150 meters, the radius of the slick is expanding by 0.1 meter/minute and its thickness is 0.08 meter. (a) At that moment, how fast is the area of the slick expanding?



    (b) The circular slick has the same thickness everywhere, and the volume of oil spilled remains fixed. How fast is the thickness of the slick decreasing when the radius is 150 meters?
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by mer1988 View Post
    Sand falls from a hopper at a rate of 0.6 cubic meters per hour and forms a conical pile beneath. Suppose the radius of the cone is always half the height of the cone.

    (a) Find the rate at which the radius of the cone increases when the radius is 2 meters.
    dr/dt = meters per hour

    (b) Find the rate at which the height of the cone increases when the radius is 2 meters.
    dh/dt = meters per hour
    a)
    V_{cone}=\frac{1}{3} \pi r^2 h
    note that h=2r and \frac{dV}{dt}=0.6 m^3/hr
    so
    V_{cone}=\frac{1}{3} \pi r^2 h=\frac{2}{3}\pi r^3
    \frac{dV}{dt}=\frac{6}{3}\pi r^2 \frac{dr}{dt} then evaluate at r=2 to solve for \frac{dr}{dt}

    b)
    h=2r, then
    \frac{dh}{dt}=2\frac{dr}{dt}

    Quote Originally Posted by mer1988 View Post
    and
    A ruptured oil tanker causes a circular oil slick on the surface of the ocean. When its radius is 150 meters, the radius of the slick is expanding by 0.1 meter/minute and its thickness is 0.08 meter.
    (a) At that moment, how fast is the area of the slick expanding?

    (b) The circular slick has the same thickness everywhere, and the volume of oil spilled remains fixed. How fast is the thickness of the slick decreasing when the radius is 150 meters?
    a)
    A_{slick}=\pi r^2
    \frac{dA}{dt}=2\pi r \frac{dr}{dt}
    use the given to solve for the rate..

    b)
    V_{slick}=\pi r^2 h
    this is just similar to the first item. the only difference is this is circular cylinder while the first one was a cone..
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