Open Top
HELP 20PTs. An open-top box with a square base is to have a volume of 108 cubic inches. Find the dimensions of?
An open-top box with a square base is to have a volume of 108 cubic inches. Find the dimensions of the box that can be made with the smallest amount of material.
Let x = side of square base
height = 108/x^2
Area = [x+ 2(108/x^2))]^2
A = [x+ 2(108/x^2))]^2
dA = 2[[x+ 2(108/x^2))](1 + 216(-2x)/x^4] = 0
[x + 216/x^2](1 – 432/x^3) = 0
x – 432/x^2 + 216/x^2 – 93312/x^5 = 0
x^6 – 432x^3 + 216x^3 – 93312 = 0
x^6 – 216x^3 – 93312 = 0
(x^3 + 216)(x^3 – 432) = 0
x^3 + 216 = 0
x^3 = -216 (discard, negative side)
(x^3 – 432) = 0
x^3 = 432
x = 432^1/3 = 7.56 inches on side
108/7.56^2 = 1.89 inch high
Box = 7.56 inch x 7.56 inch x 1.89 inch
open top
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