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Math Help - Related Rates Problem

  1. #1
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    May 2010
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    Related Rates Problem

    Hi,

    I'm currently studying for my Calc I final, and I come across a review question that I don't understand at all. Here it is:

    The value of a car V (in thousands of dollars) depends on how far it has been driven m (in thousands of miles) and its age t (in years). For a certain car it is estimated that

    V=100*\frac{(1+15e^{-m/100})}{99+t^2}

    Alex’s parents bought the car for him new a year ago but he only just recently got his license, so the car is one year old but doesn’t yet have any miles on it. If Alex starts driving it now at the rate of 10 thousand miles a year, how fast will its value decrease?


    The first problem is obviously that there are two variables in the function. I tried to get rid of that by setting

    m=10,000t

    Is that right? After that, I differentiated and ended up with a huge mess of a formula. I have a feeling that I don't even understand what the question is asking me. Can anybody help me out? Is this even a related rated problem? I don't get it at all...

    Thanks.
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  2. #2
    MHF Contributor
    Joined
    Sep 2008
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    West Malaysia
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    Quote Originally Posted by awelex View Post
    Hi,

    I'm currently studying for my Calc I final, and I come across a review question that I don't understand at all. Here it is:

    The value of a car V (in thousands of dollars) depends on how far it has been driven m (in thousands of miles) and its age t (in years). For a certain car it is estimated that

    V=100*\frac{(1+15e^{-m/100})}{99+t^2}

    Alex’s parents bought the car for him new a year ago but he only just recently got his license, so the car is one year old but doesn’t yet have any miles on it. If Alex starts driving it now at the rate of 10 thousand miles a year, how fast will its value decrease?


    The first problem is obviously that there are two variables in the function. I tried to get rid of that by setting

    m=10,000t

    Is that right? After that, I differentiated and ended up with a huge mess of a formula. I have a feeling that I don't even understand what the question is asking me. Can anybody help me out? Is this even a related rated problem? I don't get it at all...

    Thanks.
    hi

    dm/dt=10000

    dm=10000 dt

    m=\int 10000 dt

    m=10000t+c and when m=0 , t=1

    so m=10000t-10000

    Substitute this into the V equation to get rid of the m ,

     <br />
\frac{dV}{dt}=\frac{(99+t^2)(1500e^{100-100t})(-100)-(100+1500e^{100-100t})(2t)}{(99+t^2)^2}<br />

    Didn't the question say how fast the value decreases in how many years ?
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  3. #3
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    May 2010
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    Hi,

    thanks a lot, that really helps. I totally forgot to subtract 10,000.
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