Step by step
The question
Question: An open box with a square base must hold 4 m³.
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 4.
So h = 4/x².
Put it in place of h.
One x cancels: S = x² + 16 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.
16/x is 16x to the power −1, which gives −16/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 8 is 2, because 2 × 2 × 2 = 8.
6 Make sure it is the smallest
Before 2, the slope is negative: going down.
After it, positive: going up.
Down, then up: the bottom.
A minimum.
7 Answer the question
The height: 4/4 is 1.
So the least material is 12 m²: a base 2 by 2, 1 high.
8 Check
Let's check with x a bit smaller: 1 gives 17.
And x = 3 gives 14.33.
Both are bigger than 12.
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.