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Math Help - [SOLVED] minimums

  1. #1
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    [SOLVED] minimums

    A traveller employs a man to drive him from sydney to Wollongong for a hourly payment of P dollars. Running costs of the car, which are also paid by the traveller are kv^3 dollars per hour, where vkn/h is the speed and k is a constant. Find the uniform speed that will minimize the total cost.

    i don't get it T_T
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  2. #2
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    Hello, chibiusagi!

    We're having trouble reading what you wrote.
    Let me take a guess at what you meant . . .


    A traveller employs a driver from Sydney to Wollongong at P dollars per hour.
    Operating cost of the car is kv^3 dollars per hour,
    where v km/hr is the speed and k is a constant.
    Find the uniform speed that will minimize the total cost.

    Let H = number of hours of driving.

    The driver is paid P dollars per hour for H hours.
    . . This will cost: . PH dollars.

    The car will cost kv^3 dollars per hour for H hours.
    . . This will cost: . kv^3H dollars.

    The total cost is: . C \;=\;PH + kv^3H dollars. .[1]


    Let D = distance from Sydney to Wollongong (km).
    . . At v km/hr, it will take: . H \:=\:\frac{D}{v} hours. .[2]


    Substitute [2] into [1]: . C \;=\;kv^3\left(\frac{D}{v}\right) + P\left(\frac{D}{v}\right) \;=\;kDv^2 + PDv^{-1}

    Minimize: . C' \;=\;2kDv - PDv^{-2} \:=\:0

    Multiply by v^2\!:\;\;2kDv^3 - PD \:=\:0 \quad\Rightarrow\quad 2kDv^3 \:=\:PD\quad\Rightarrow\quad v^3 \:=\:\frac{P}{2k}

    Therefore: . v \;=\;\sqrt[3]{\frac{P}{2k}}

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