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.
:
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/x^2}}}
:
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/x^2}}}}
S.A. = x^2 + 4(x*{{{1000/x^2}}})
cancel x into x^2
S.A. = x^2 + {{{4000/x}}}
Graph this in your graphing calc S.A. = y
{{{ graph( 300, 200, -5, 20, -200, 1000, x^2+(4000/x)) }}}
minimum surface when x = 12.6 cm the side of the square base
Find the height
h = {{{1000/12.6^2}}}
h = {{{1000/158.76}}}
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