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Math Help - Help with an inverse function story problem please

  1. #1
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    Help with an inverse function story problem please

    A trough has a semicircular cross section with a radius of 3 feet. Water starts flowing into the trough in such a way that the depth of the water is increasing at a rate of 2 inches per hour.

    Help with an inverse function story problem please-aba97104b52d98f706b0e95cbcf7fa.gif

    a. Give a function w = f(t) relating the width w, in feet of the surface of the water to the time t, in hours.
    b.
    After how many hours will the surface of the water have width of 4 feet? (Round your answer to two decimal places.)

    I am really confused on how to approach these two questions...Any help is appreciated!
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Help with an inverse function story problem please

    I would begin by drawing a line segment from the center of the semi-circle to the edge of the water, and a vertical line from the center of the semi-circle to the water. You now have a right triangle whose legs are w/2 and 3 - h (where h is the depth of the water), and the hypotenuse is 3. Use the Pythagorean theorem and solve for w.

    Once you have this, which is w(h), you want to express h as a function of t. Can you proceed now?
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  3. #3
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    Re: Help with an inverse function story problem please

    Hey UWM120.

    Once you get an expression for the volume at a particular point in time, you can use the volume rotation integral to solve for a particular value of t where that t will give you the height of the spherical bowl and from that you can find the width by intersecting a line (or a plane, doesn't matter) with the sphere itself.

    The volume rotation formula is the integral of pi*[f(x)]^2dx rotated around the x-axis and your limits will be from u to r where you have the volume already (given by V(t)) and you need to solve for a particular value of u. Recall that the equation for a circle is y^2 = r^2 - x^2 = [f(x)]^2 so you can evaluate this integral using standard anti-derivative laws.
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