Step by step
The question
Question: A rectangular garden of 50 m² is fenced on three sides.
A wall is the fourth side.
Find the least fencing needed.
1 Draw and name
The wall is one side: no fence there.
The two sides that touch the wall 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 50.
So y = 50 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.
50/x is 50x to the power −1, which gives −50/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 25, so x is 5.
Minus 5 squared is also 25, but x must be positive.
Careful!
A length cannot be negative.
Only x = 5 is allowed.
6 Make sure it is the smallest
Before 5, 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: 50/5 is 10.
So the least fencing is 20 m.
8 Check
Let's check with x a bit smaller: 4 gives 20.5.
And x = 6 gives 20.33.
Both are bigger than 20.
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.