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Math Help - Time rate of change of cost problem

  1. #1
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    Time rate of change of cost problem

    The cost of manufacturing x cases of umbrellas is C dollars (in thousands), where

    Monthly production at t weeks is expected to be

    I know that = 18(5000+80t)+4sqrt(5000+80t)+9

    and that marginal cost or = 28+(2/x^.5)

    I'm not sure how to find in order to solve the equation for t=2 in order to find how fast costs are rising per week.
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  2. #2
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    Quote Originally Posted by terrytriangle View Post
    The cost of manufacturing x cases of umbrellas is C dollars (in thousands), where

    Monthly production at t weeks is expected to be

    I know that = 18(5000+80t)+4sqrt(5000+80t)+9

    and that marginal cost or = 28+(2/x^.5)

    I'm not sure how to find in order to solve the equation for t=2 in order to find how fast costs are rising per week.
    Hiya Terry by the step where you list = 18(5000+80t)+4sqrt(5000+80t)+9 , you are on the right track. Since you now have C(t), you can differentiate to find  \frac{dC}{dt} without having to worry about the chain rule with other variables.

    The derivative of = 18(5000+80t)+4sqrt(5000+80t)+9 would be  \frac{dC}{dt} = 160*(5000+80t)^(-1/2) + 1440t. C'(2)=160*(5000+80(2))^(-1/2) + 1440(2)

    To take the derivative of the square root of 5000+80t a chain rule must be used so
     \frac{d}{dt} (5000+80t)^(1/2)=

     \frac{1}{2} (5000+80t)^(-1/2) * 80 since 80 is the derivate of the inside term.
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