Question 1028760
A rectangular box is to have a square base and a volume of 40 ft3.
 If the material for the base costs $0.35/ft2, the material for the sides costs $0.05/ft2, and the material for the top costs $0.15/ft2, determine the dimensions of the box that can be constructed at minimum cost.
:
let x = the length of the side of the square base
then
{{{40/x^2}}} = the height of the box
:
Area of the top and bottom will be x^2
Area of each of the 4 sides will be :
{{{x(40/x^2)}}} simplifies to {{{40/x}}}
Fours sides area: 4*{{{40/x}}} = {{{160/x}}}
:
The cost equation
Cost = base cost + side cost + top cost
C(x) = {{{.35x^2 + .05(160/x) + .15x^2}}}
C(x) = {{{.50x^2 + 8/x}}}
Graph this equation to find the min cost
{{{ graph( 300, 200, -2, 5, -4, 20, .5x^2+(8/x)) }}}
minimum cots occurs when x = 1.7 ft, the length and width of the box
Find the height
h = {{{40/1.7^2}}}
h = 13.8 ft
:
The box dimension for min cost: 1.7 by 1.7 by 13.8