SOLUTION: An open rectangular box with square base and open top is to contain 1000cm^3.Find the dimensions that require the least amount of material.Neglect the thickness of the material and

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Question 962250: An open rectangular box with square base and open top is to contain 1000cm^3.Find the dimensions that require the least amount of material.Neglect the thickness of the material and waste in construction.(Hint:Here we are looking at surface area)
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Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
An open rectangular box with square base and open top is to contain 1000cm^3.
Find the dimensions that require the least amount of material.
:
let x = side of the square base
let h = the height of the box
then the volume
x * x * h = 1000
x^2h = 1000
h = 1000%2Fx%5E2
:
The surface area of an open box
:
S.A. = bottom area + 4 side areas
S.A. = x^2 + 4(x*h)
Replace h with 1000%2Fx%5E2}
S.A. = x^2 + 4(x*1000%2Fx%5E2)
cancel x into x^2
S.A. = x^2 + 4000%2Fx
Graph this in your graphing calc S.A. = y
+graph%28+300%2C+200%2C+-5%2C+20%2C+-200%2C+1000%2C+x%5E2%2B%284000%2Fx%29%29+
minimum surface when x = 12.6 cm the side of the square base
Find the height
h = 1000%2F12.6%5E2
h = 1000%2F158.76
h = 6.3 cm is the height
:
Summarize, 12.6 by 12.6 by 6.3 dimensions for minimum surface area
:
confirm this by finding the volume with these dimension
12.6 * 12.6 * 6.3 = 1000.2, close enough