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Thread: optimization

  1. #1
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    Question optimization

    A cylindrical can, open at the top, is to hold 1000cm^3 of liquid. The material for the side of the can costs $.50 per cm^2 to manufacture, and the material for the base costs $.75 per cm^2 to manufacture. Find the height and radius that minimize the cost to manufacture the can.

    Does anyone know how to solve this problem? I'm a bit lost.
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  2. #2
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    Quote Originally Posted by bluebyte22 View Post
    A cylindrical can, open at the top, is to hold 1000cm^3 of liquid. The material for the side of the can costs $.50 per cm^2 to manufacture, and the material for the base costs $.75 per cm^2 to manufacture. Find the height and radius that minimize the cost to manufacture the can.

    Does anyone know how to solve this problem? I'm a bit lost.
    volume information ...

    $\displaystyle \pi r^2 h = 1000$

    $\displaystyle h = \frac{1000}{\pi r^2}$

    surface area ...

    $\displaystyle A = \pi r^2 + 2\pi rh$

    $\displaystyle A = \pi r^2 + 2\pi r \left(\frac{1000}{\pi r^2}\right)$

    cost ...

    $\displaystyle C = .75(\pi r^2) + .50\left[2\pi r \left(\frac{1000}{\pi r^2}\right)\right]$

    clean up the algebra for the cost function, find $\displaystyle \frac{dC}{dr}$ and minimize the cost.
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  3. #3
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    Red face

    Thank you
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