Step by step
The question
Question: A farmer has 80 m of fence to make a rectangle next to a river.
Find the largest area.
1 Name the unknown
The river is one side: no fence there.
Call each side that touches the river x.
The fence makes three sides: x, x, and the side across.
Together they are 80 m.
So the side across is 80 − 2x.
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) = 80x − 2x².
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) = 80 − 4x.
Set it equal to 0.
Add 4x to both sides, so they stay equal.
To remove times 4, we divide by 4.
80 divided by 4 is 20.
4 Is it the biggest?
Before 20 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 40.
So the largest area is 800 m².
6 Check
Let's check with x a bit smaller: 19 gives 798.
And x = 21 gives 798.
Both are smaller than 800.
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.