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

Answer 1

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.