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Math Help - Optimization

  1. #1
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    Optimization

    1. The "gait" of an animal is a measure of how jerky or smooth the animal's motion appers when it is running. Gait, g can be shown to be related to the power, P, necessary for the animal to run at a given speed. For an animal 1m long, running at a velocity of 10m/s, the power is given by
    P= 0.1g + 1000/1+g.
    Determine the gait that minimizes the required power for the animal to run.

    2. Jeff is making a toy box for his nephew. He will use pine for the square base and sides. but oak for the top. Pine costs $16/m squared and oak costs $20/m squared. If the volume of the toy box is to be 1.3 m cubed, fined the amount of oak that is needed if the cost of all the wood for the toy box is to be as low as possible.
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  2. #2
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    Quote Originally Posted by aaasssaaa View Post
    1. The "gait" of an animal is a measure of how jerky or smooth the animal's motion appers when it is running. Gait, g can be shown to be related to the power, P, necessary for the animal to run at a given speed. For an animal 1m long, running at a velocity of 10m/s, the power is given by
    P= 0.1g + 1000/1+g.
    Determine the gait that minimizes the required power for the animal to run.
    I presume that the formula is:
    P = 0.1g + 1000/(1 + g)
    PLEASE use parenthesis!!

    To find the minimum we take the first derivative:
    dP/dg = 0.1 + 1000*(-1)/(1 + g)^2

    then set it equal to 0:
    0.1 + 1000*(-1)/(1 + g)^2 = 0

    0.1 = 1000/(1 + g)^2

    0.1(1 + g)^2 = 1000

    (1 + g)^2 = 1000/0.1 = 10000

    1 + g = 100 <-- Technically (+/-)100, but since g can't be negative...

    g = 100 - 1 = 99

    I'll let you prove that this value of g provides a minimum.

    -Dan
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