Step by step
The question
Question: An open box with a square base must hold 256 in³.
Find the least material needed.
1 Draw and name
The base is a square: x by x.
The height is h.
There is no top.
2 Write the material
The material is the base, x squared, plus 4 sides, each x by h.
3 Use only one letter
S has two letters.
The volume gives the link: base times height is 256.
So h = 256/x².
Put it in place of h.
One x cancels: S = x² + 1024 over x.
4 Which x make sense
A length is positive: x is more than 0.
5 Slope = 0
The smallest value is where the slope is 0.
x² gives 2x.
1024/x is 1024x to the power −1, which gives −1024/x².
Set it to 0, and move the fraction across.
Times both sides by x², so they stay equal.
Divide by 2.
The cube root of 512 is 8, because 8 × 8 × 8 = 512.
6 Make sure it is the smallest
Before 8, the slope is negative: going down.
After it, positive: going up.
Down, then up: the bottom.
A minimum.
7 Answer the question
The height: 256/64 is 4.
So the least material is 192 in²: a base 8 by 8, 4 high.
8 Check
Let's check with x a bit smaller: 7 gives 195.29.
And x = 9 gives 194.78.
Both are bigger than 192.
Right!
Let's look back.
First: draw and name.
Then: write the material.
Then: use only one letter.
Then: which x make sense.
Then: slope = 0.
Then: make sure it is the smallest.
Then: answer the question.
Last: check.