Question 253460
An open top box with a square base is to be constructed from a sheet metal in such a way that the completed box is made of 2m^2 of the sheet metal.
 Express the volume of the box as a function of the width of the base.
:
The side of a square piece of sheet metal of 2 sq/m = {{{sqrt(2)}}} m
:
Let x = side of the square base of the box
:
let h = side of the 4 small squares removed from the square sheet to make the box
Also h = the height of the open box
:
x = {{{sqrt(2) - 2h}}}
solve in terms of h
2h = {{{sqrt(2) - x}}}
:
h = {{{(sqrt(2) - x)/2}}}
:
Vol = x^2 * h
replace h with {{{(sqrt(2) - x)/2}}}
:
V(x) = {{{x^2 * ((sqrt(2) - x)/2)}}}
:
:
Max vol occurs when x = .9 m, find h
h = {{{(sqrt(2) - .9)/2}}}
h = .26 m
:
.9^2 * .26 = .21 cu meters