base of box: x by x inches
height of box : y inches
V = x^2 y
1600 = x^2 y
y = 1600/x^2
cost = 3(2x^2) + 5(4xy)
= 6x^2 + 20x(1600/x^2) = 6x^2 + 32000/x
d(cost)/dx = 12x - 32000/x^2
= 0 for a min of cost
12x = 32000/x^2
x^3 = 8000/3
x = 20/3^(1/3) = appr 13.867
sub that into y = 1600/x^2
y = appr 8.320
state your conclusion
Melissa wants to make a rectangular box with a square base and cover its top and bottom faces with velvet, which will cost her $3 per square inch, and the sides with silk, which will cost her $5 per square inch. The box should have a volume of 1600 cubic inches. Find the dimensions of the box that will cost her the least amount of money.
1 answer