Step by step
The question
Question: A rectangle has a perimeter of 40 m.
Find the largest area.
1 Name the unknown
Call the width x.
The perimeter goes round all 4 sides: width and length, twice.
So width plus length is half of 40: 20.
So the length is 20 − x.
2 Write the area
Area is one side times the other.
The x in front of the bracket multiplies every part inside.
Now we have A(x) = 20x − x².
3 Solve f′(x) = 0
The biggest area is at the top of the curve, where the slope is 0.
The power rule gives A′(x) = 20 − 2x.
Set it equal to 0.
Add 2x to both sides, so they stay equal.
To remove times 2, we divide by 2.
20 divided by 2 is 10.
4 Is it the biggest?
Before 10 the slope is positive: going up.
After it, negative: going down.
Up, then down: a maximum.
Careful!
x is only one side.
The question asks for the area.
5 Answer the question
The other side is 10.
So the largest area is 100 m².
6 Check
Let's check with x a bit smaller: 9 gives 99.
And x = 11 gives 99.
Both are smaller than 100.
Right!
Let's look back.
First: name the unknown.
Then: write the area.
Then: solve f′(x) = 0.
Then: is it the biggest?.
Then: answer the question.
Last: check.