A box of volume 72 m3 with a square bottom and no top is constructed out of two different materials. the cost of the bottom is $40/m2 and the cost of the sides is $30/m2. find the dimensions of the box that minimize total cost.

Square bottom, so L=W

yah

V=HL²

surface area (minus the top) is

SA=2H (2L) + LW

SA=4HL+L²

great

given V=72

72=HL²

solve for H

72 / (L²) = H

umm

sorry, I'm all over the place

cost=30 (4) HL+40L² or

cost=120HL+40L²

subsitute for H

cost=120 (72 / (L²)) L+40L²

cost=8640/L+40L²

find the minimum using your calculator or take the derivitive

min at L=3∛4≈4.7622031559046

anyway, sub back

72/L²=H

72 / ((3∛4) ²) = H

2∛4=H

the dimentions of the box are

height is 2∛4 or about 3.1748m

length and width are 3∛4 or about 4.7622m