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calculus

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A rectangular box is to be constructed from two different materials. The box will have a square base and open top. The material for the bottom costs $4.25/m2. The material for the sides costs $2.50/m2. Find the dimensions of the box with the largest volume if the budget is $500 for the material

  • calculus - ,

    Volume=a^2 h
    cost= 4.25*a^2 + 2.5*(4 a*h)

    dV/da=2ah+a^2 dh/da=0
    but on the cost equation
    take the derivative of Cost respect to a

    0=8.5a+10h+ 10a*dh/da

    but in the volume equation we can find
    dh/da=-2ah/a^2=-2h/a

    put that into the derivative of the cost
    0=8.5a+10h+10 a(-2h/a)

    0=8.5a-10h
    h=.85a
    now you can put that into the cost equation, with cost at 500, and solve for a, then h.

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