Question 1128128
A rectangular storage container with an open top is to have a volume of 18 cubic meters.
 The length of it's base is twice the width.
:
let x = the width of the base
then
2x = the length of the base
and
h = height of the box
:
2x^2 = the area of the base
2(2xh) = 4xh = area of the two vertical sides
2xh = area of the two other vertical sides
then
4xh + 2xh = 6xh, the area of the 4 vertical sides

The volume equation
{{{2x^2*h = 18}}}
h = {{{18/(2x^2)}}}
h = {{{9/x^2}}}
:
 Material for the base costs 10 dollars per square meter.
 Material for the sides costs 8 dollars per square meter.
 Find the cost of materials for the cheapest such container.
Cost equation
C = 10(2x^2) + 8(6xh)
C = 20x^2 + 48xh
Replace h with {{{18/(2x^2)}}}
C(x) = 20x^2 + 48x({{{9/(x^2)}}})
Cancel x
C(x) = 20x^2 + 48({{{9/x}}})
C(x) = 20x^2 + ({{{432/x}}})
:
{{{ graph( 300, 200, -2, 4, -100, 400, (20x^2)+(432/x) ) }}} 
minimum cost:
 x = 2.25 meters is the width of the base
and
2(2.25) = 4.5 meters is the length of the base
:
find the height
 h = {{{9/(2.25^2)}}}
 h = {{{9/5.0625}}}
 h = 1.78 meters is the height
:
cost of the open box
C = 10(4.5*2.25) + 8(6*2.25*1.78)
C = 10(10.125) + 8(24)
C = 101.25 + 192
C = $293.25 the min cost for an open box which is: 2.25 by 4.5 by 1.78 m