Question 428043
The box is open, so it will have 5 sides.
Let a = the side length of the square bottom
Let h = the height of the box
We are given the volume of the box = 5000.
We need to find a formula for the amount of material required for the box.
The bottom will have area {{{a^2}}}
Each side will have area {{{ah}}}, and there are 4 of them.
Thus, the total amount of material will be:
{{{M = a^2 + 4ah}}} (1)
The volume of the box, {{{a^2h = 5000}}}
So {{{h = 5000/a^2}}}
Substitute this value for h into equation (1):
{{{M = a^2 + 4a(5000/a^2) = a^2 + 20000/a}}}
To minimize M, we take the derivative and set = 0:
{{{0 = 2a - 20000/a^2}}}
Solve for a:
{{{a^3 = 10000 -> a = 21.544}}} m
Therefore {{{h = 5000/21.544^2 = 10.772}}} m