calculus
posted by j .
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

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=8.5a10h
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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