Step by step
The question
Question: A rectangular pen of 72 ft² is fenced on three sides.
A barn is the fourth side.
Find the least fencing needed.
1 Draw and name
The barn is one side: no fence there.
The two sides that touch the barn are x.
The side across is y.
2 Write the fencing
The fence is two x sides and one y side.
3 Use only one letter
Two letters.
The area gives the link: x times y is 72.
So y = 72 over x.
Put it in place of y.
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.
2x gives 2.
72/x is 72x to the power −1, which gives −72/x².
Set it to 0, and move the fraction across.
Times both sides by x², so they stay equal.
Divide by 2.
x squared is 36, so x is 6.
Minus 6 squared is also 36, but x must be positive.
Careful!
A length cannot be negative.
Only x = 6 is allowed.
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 other side: 72/6 is 12.
So the least fencing is 24 ft.
8 Check
Let's check with x a bit smaller: 5 gives 24.4.
And x = 7 gives 24.29.
Both are bigger than 24.
Right!
Let's look back.
First: draw and name.
Then: write the fencing.
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.