Step by step
The question
Question: A farmer has 60 m of fence to make a rectangle next to a wall.
Find the largest area.
1 Name the unknown
The wall is one side: no fence there.
Call each side that touches the wall x.
The fence makes three sides: x, x, and the side across.
Together they are 60 m.
So the side across is 60 − 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) = 60x − 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) = 60 − 4x.
Set it equal to 0.
Add 4x to both sides, so they stay equal.
To remove times 4, we divide by 4.
60 divided by 4 is 15.
4 Is it the biggest?
Before 15 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 30.
So the largest area is 450 m².
6 Check
Let's check with x a bit smaller: 14 gives 448.
And x = 16 gives 448.
Both are smaller than 450.
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.