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Math Help - Functions of fuel consumption

  1. #1
    Senior Member I-Think's Avatar
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    Functions of fuel consumption

    For a truck, its fuel consumption in \frac{ml}{hr} is given by
    f(v)=(v-10)^2+9900, where v is the speed in \frac{km}{hr}. Find the speed that gives the best distance to fuel ratio

    No calculus is to be used to solve this question

    Any pointers? Was thinking of trying \frac{f(v)}{v}, but didn't know how to find the minimum without calculus
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    Re: Functions of fuel consumption

    Quote Originally Posted by I-Think View Post
    For a truck, its fuel consumption in \frac{ml}{hr} is given by
    f(v)=(v-10)^2+9900, where v is the speed in \frac{km}{hr}. Find the speed that gives the best distance to fuel ratio

    No calculus is to be used to solve this question

    Any pointers? Was thinking of trying \frac{f(v)}{v}, but didn't know how to find the minimum without calculus
    Hi I-Think,

    To find the best distance to fuel ratio we should find the speed that minimizes f(v). f(v) will be minimum when (v-10)^2 is a minimum. It is clear that the minimum value of (v-10)^2 is zero. Hence the corresponding speed is,

    (v-10)^2=0

    v=10\mbox{ kmh^{-1}}
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  3. #3
    Senior Member I-Think's Avatar
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    Re: Functions of fuel consumption

    Are you sure? Talked to a friend and he said the answer was 100. He didn't give reasons, just hints.

    Checking at v=10, we have f(v)=9900, so that is for every 10km traveled in an hour, 9900ml of fuel is consumed, giving us a consumption rate of 990mlkm^{-1}

    But at v=100, we have f(v)=8100+9900=18000, that is for every 100km traveled in an hour, 18000ml of fuel is consumed, giving us a consumption rate of 180mlkm^{-1}

    Is there something wrong with this reasoning?
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    Re: Functions of fuel consumption

    Quote Originally Posted by I-Think View Post
    Are you sure? Talked to a friend and he said the answer was 100. He didn't give reasons, just hints.

    Checking at v=10, we have f(v)=9900, so that is for every 10km traveled in an hour, 9900ml of fuel is consumed, giving us a consumption rate of 990mlkm^{-1}

    But at v=100, we have f(v)=8100+9900=18000, that is for every 100km traveled in an hour, 18000ml of fuel is consumed, giving us a consumption rate of 180mlkm^{-1}

    Is there something wrong with this reasoning?
    No, there isn't. My method is incorrect. I had found the speed for which the fuel consumption is minimum, not the speed corresponding to the best distance to fuel ratio. As you have correctly stated we have to find the minimum of \frac{f(v)}{v}. But alas I still didn't get an idea to tackle this problem.
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