Step by step
The question
Question: A rectangle has an area of 64 ft².
Find the smallest perimeter.
1 Draw and name
Call the sides x and y.
2 Write the perimeter
The perimeter is two x sides and two y sides.
3 Use only one letter
Two letters.
The area gives the link: x times y is 64.
So y = 64 over x.
Put it in place of y.
Now we have P = 2x + 128 over x.
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.
128/x is 128x to the power −1, which gives −128/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 64, so x is 8.
Minus 8 squared is also 64, but x must be positive.
Careful!
A length cannot be negative.
Only x = 8 is allowed.
6 Make sure it is the smallest
Before 8, 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: 64/8 is 8.
So the smallest perimeter is 32 ft.
The rectangle is a square!
8 Check
Let's check with x a bit smaller: 7 gives 32.29.
And x = 9 gives 32.22.
Both are bigger than 32.
Right!
Let's look back.
First: draw and name.
Then: write the perimeter.
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.