A rectangular box with a volume of 960 ftcubed3 is to be constructed with a square base and top. The cost per square foot for the bottom is 15cents¢ , for the top is 10cents¢ , and for the sides is 1.5cents¢. What dimensions will minimize the cost?

Yas

Square base and top, means L=W

so

V=HL²

V=960

bottom cost=15L²

top cost=10L²

sides=2H (L+W) 1.5 or 3H (2L) or 6HL

so

total cost=15L²+10L²+6HL or

TC=25L²+6HL

eliminate height

we know

960=HL²

divide both sides by L²

960/L²=H

sub that for H

TC=25L²+6 (960/L²) L

TC=25L²+5760/L

find where TC is minimum

take the derivitive

50L-5760/L²

or

(10) (5L³-576) / (L²)

set numerator to zero

5L³-576=0

solve

L=0.8∛225

H=960/L²

H = (20/3) ∛225

if we were to sub, we would find that minimum cost is 720 (∛15) or aprox 1775.67 cents or $17.76

dimentions

L=0.8∛225 aprox 4.86576 ft

W=0.8∛225 aprox 4.86576 ft

H = (20/3) ∛225 aprox 40.548 ft

min cost is $17.76