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Math Help - Optimization problem

  1. #1
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    Optimization problem

    A small business uses a minivan to make deliveries. The cost per hour for fuel is F=(v^2)/(360), where v is the speed of the minivan (in miles per hour). The driver is paid $10 per hour. Find the speed that minimizes the cost of a 110-mile trip. (Assume there are no costs other than fuel and wages.)

    I don't know how to set this particular problem up
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  2. #2
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    Quote Originally Posted by dassix View Post
    A small business uses a minivan to make deliveries. The cost per hour for fuel is F=(v^2)/(360), where v is the speed of the minivan (in miles per hour). The driver is paid $10 per hour. Find the speed that minimizes the cost of a 110-mile trip. (Assume there are no costs other than fuel and wages.)

    I don't know how to set this particular problem up
    1. Costs for the fuel: F(t)=\frac1{360} v^2 \cdot t

    Wages of the driver w(t)=10 t. Therefore: The total costs are:

    c= \frac1{360} v^2 \cdot t + 10t

    2. From the definition of speed you'll get:

    speed=\frac{distance}{time} \text{  that means:  } v=\frac{110}t~\implies~t=\frac{110}v

    3. Plug in this term into the equation of the total costs:

    c(v)=\frac{110}{360} v+\frac{1100}v

    4. Differentiate c(v) wrt v and solve for v the equation c'(v) = 0.
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