Question 1099943
A box is constructed out of two different types of metal.
 The metal for the top and bottom, which are both square, costs $5 per square foot and the metal for the sides costs $2 per square foot.
 Find the dimensions that minimize cost if the box has a volume of 15 cubic feet.
:
let s = one side of the square base
let h = the height of the box
then
s^2 * h = 15 cu/ft
Therefore
h = {{{15/s^2}}}
:
surface area = top, bottom, 4 side areas
SA = 2s^2 + 4hs
replace h with {{{15/s^2}}}
SA = 2s^2 + 4s*{{{15/s^2}}}
cancel the s
SA = 2s^2 + (60/s)
Cost of the box
C(x) = 5(2x^2) + 2(60/s)
C(x) = 10x^2 + {{{120/s}}}
:
Graphically, y axis = the cost
{{{ graph( 300, 200, -3, 5, -20, 160, 10x^2+(120/x), 99.26) }}}
minimum cost when x=1.9 ft
find the height
h = {{{15/1.9^2}}}
h = 4.155 ft
:
The dimensions for minimum cost: 1.9 by 1.9 by 4.155 ft Cost about $99.26 (green)