Step by step
The question
Question: An open box with a square base must hold 108 cm³.
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 108.
So h = 108/x².
Put it in place of h.
One x cancels: S = x² + 432 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.
432/x is 432x to the power −1, which gives −432/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 216 is 6, because 6 × 6 × 6 = 216.
6 Make sure it is the smallest
Before 6, the slope is negative: going down.
After it, positive: going up.
Down, then up: the bottom.
A minimum.
7 Answer the question
The height: 108/36 is 3.
So the least material is 108 cm²: a base 6 by 6, 3 high.
8 Check
Let's check with x a bit smaller: 5 gives 111.4.
And x = 7 gives 110.71.
Both are bigger than 108.
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.