Question

A box with a square and open top must have a volume of 4000cm^3. Find the dimensions of the box that minimizes the amount of materials used.

Answer 1

remember @esam2 that the formula for volume of a rectangular prism is as follows:

\(\Huge{V}=\) \(\Huge{w}\times{h}\times{l}\)

Answer 2

V represents volume, w represents weight, h represents height, and l represents length

Answer 3

Yes, I remember .
w=width

Answer 4

yes :) so if you keep these numbers as even as possible, then you will lessen the amount of materials needed. So we're here to figure out the dimensions to make this possible

Answer 5

okay

Answer 6

So this is an optimization problem :)

let the side of the square base=s
Let the height of the box = h
Then the volume of the box, V=s^2h

The area of the box is s^2
There are 4 sides, and each side has an area of sh
So total area of 4 sides is equivalent to 4sh

If you add everything that we've found already, 4 sides + base=s^2 + 4sh.
you following me so far? @esam2 ?

Answer 7

I think so..

Answer 8

alright.
Let A = s^2 + 4sh

So in the problem, we're told that the volume is 4000 cm^3, which we established equals s^2h, and A = s^2 + 4sh
Therefore, h= 4000 ÷ s^2 : so A= s^2 + 4(4000÷ s^2) = s^2 + 16,000 ÷ s

Therefore, h= 4000 divided by s^2: so the area equals s^2 + 4( 4000/s^2) which equals s^2 + 16,000 divided by s

***i.e. A= s^2 +16,000÷ s

Answer 9

yikes, the glitch is real. Basically, you should be able to find that the area is equivalent to s^2 +16,000/s

\[A= s ^{2} + 16,000\div s\]

^Wrote it out more clearly ^_^

Answer 10

using this, we can find the minimum area

Answer 11

Oh okay, I get it! Thanks for your help! :)