SOLUTION: A rectangular storage container with an open top is to have a volume of 18 cubic meters. The length of its base is twice the width. Material for the base costs 10 dollars per squa

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Question 1128128: A rectangular storage container with an open top is to have a volume of 18 cubic meters. The length of its base is twice the width. 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.

Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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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%5E2%2Ah+=+18
h = 18%2F%282x%5E2%29
h = 9%2Fx%5E2
:
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%2F%282x%5E2%29
C(x) = 20x^2 + 48x(9%2F%28x%5E2%29)
Cancel x
C(x) = 20x^2 + 48(9%2Fx)
C(x) = 20x^2 + (432%2Fx)
:
+graph%28+300%2C+200%2C+-2%2C+4%2C+-100%2C+400%2C+%2820x%5E2%29%2B%28432%2Fx%29+%29+
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%2F%282.25%5E2%29
h = 9%2F5.0625
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