Find the minimum cost of a rectangular box

Question

Find the minimum cost of a rectangular box of volume 150 cm³ whose top and bottom cost 4 cents per cm² and whose sides cost 6 cents per cm².

 

cost= ? cents

Answer 1

rectangular box 150 cubic centimeters

sides cost 6 cents per square cm

top & bottom cost 4 cents per cm^2

what is the minimum cost to make the box? about $9.39

sides = x and y

height = h

volume = xyh = 150, XY = 150/h

top & bottom areas = XY each

4 side areas = 2xh + 2yh

cost of top and bottom = 4xy times 2= 8xy

cost of 4 sides = 6(2xh+2yh) = 12xh + 12yh

total cost = C = 8xy + 12h(x+y) = 8(150)/h + 12h(x+y)

C'(h) = -1200/h^2 + 12x+12y = 0

1200 = (12x+12y)h^2

100 = (x+y)h^2

h^2 = 100/(x+y)

h = 10/sqr(x+y) Let x=y

h = 10/sqr(2x) = (10/sqr2)/sqrx

h= 5sqr2/sqrx = 5(sqr(2/x))= 150/x^2

sqr(2/x) = 30/x^2

2/x = 900/x^4

2 = 900/x^3

x^3 = 900/2 = 450

x = cube root of 450 = about 7.663 cm = y

h = 150/XY = 150/7.663^2 = about 2.554

minimum cost = 4(2)XY + 6(2xh +2yh) = 8(7.663^2) + 12(2.554)(2)(7.663) = 469.77 + 469.71= 939.42 cents

=about $9.39