SOLUTION: A closed box is to be a rectangular solid with a square base and a volume of 12 cubic feet. Find the most economical dimensions if the top of the box is twice as expensive as the s

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Question 241084: A closed box is to be a rectangular solid with a square base and a volume of 12 cubic feet. Find the most economical dimensions if the top of the box is twice as expensive as the sides and the bottom?
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
A closed box is to be a rectangular solid with a square base and a volume of 12 cubic feet.
Find the most economical dimensions if the top of the box is twice as expensive as the sides and the bottom?
:
Let dimension of the square bottom = x, area of the bottom and top = x^2
volume
let h = the height of the box
x^2 * h = 12
h = 12%2Fx%5E2
:
Surface area,
Since the top cost twice as much as the bottom, we can treat them like 3x^2 instead of 2x^2,
:
S.A. = 3x^2 + 4(x*h)
Replace h with 12%2Fx%5E2
SA = 3x^2 + 4x*12%2Fx%5E2
Cancel an x
SA = 3x^2 + 48%2Fx
:
Plot this equation
+graph%28+300%2C+200%2C+-2%2C+4%2C+-10%2C+60%2C+3x%5E2%2B%2848%2Fx%29%29+
:
Minimum surface area is minimum expense, occurs when x = 2 inches
:
Find h:
h = 12%2Fx%5E2
h = 12%2F2%5E2
h = 3 inches
:
2 by 2 by 3 is the dimensions of the box
:
You can see that if the top and bottom cost the same we would have
2x^2 + 48%2Fx and the graph would be:
+graph%28+300%2C+200%2C+-2%2C+4%2C+-10%2C+60%2C+2x%5E2%2B%2848%2Fx%29%29+
about x = 2.25 inches
:
:
HOpe this made sense to you