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Math Help - Function Problem

  1. #1
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    Function Problem

    A street light is 12 feet above a straight bike path. Olav is bicycling down the path at a rate of 15 MPH. At midnight, Olav is 39 feet from the point on the bike path directly below the street light. (See the picture.) The relationship between the intensity C of light (in candlepower) and the distance d (in feet) from the light source is given by C = (k/d^2), where k is a constant depending on the light source.

    Function Problem-64ff1e02cf0d8fdcfadf1e8b9733e0.gif

    a) From22 feet away, the street light has an intensity of 1 candle. What is k?
    22^2 = k = 484

    b)
    Find a function which gives the intensity I of the light shining on Olav as a function of time t, in seconds.

    So below is the answer for "b" but I am confused on how to get this, can someone explain to me? Thank you!
    Last edited by UWM120; November 11th 2012 at 09:02 PM.
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Function Problem

    From the result of part a), we know:

    I=\frac{22^2}{d^2}

    Using the Pythagorean theorem, we may state:

    d^2=12^2+x^2

    where x is the distance from the point directly below the lamp that Olav is at at time t.

    If midnight is set to time t=0, then we have:

    \frac{dx}{dt}=15\cdot\frac{\text{mi}}{\text{hr}} \cdot\frac{5280\text{ ft}}{1\text{ mi}} \cdot\frac{1\text{ hr}}{3600\text{ s}}=22\frac{\text{ft}}{\text{s}} where x(0)=-39

    Thus:

    x(t)=22t-39 and so:

    d^2=12^2+(22t-39)^2=484t^2-1716t+1665

    and thus, we find:

    I(t)=\frac{484}{484t^2-1716t+1665}
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  3. #3
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    Re: Function Problem

    Thank you so much!
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