What's known:

A rectangle with perimeter of 20 m.

The question:

Express the area as a function of length.

Answer:

We can call the length of the rectangle as l, and the breadth of the rectangle as b, and then,

Perimeter = 2 × (length + breadth)

20 = 2 × (l + b)

20 = 2l + 2b // Divide both sides by 2, and you will have,

10 = l + b

or,

l + b = 10 (first equation)

And then the Area.

Area = length × breadth

A = l × b (second equation)

We want the Area to be described as a function of length, so we don't want that 'b' in the formula.

So what we can do is redefine the 'b' based on the first equation.

l + b = 10

b = 10 - l

And after that, we can subtitute the new value of b (that is 10 - l) to the second equation.

A = l × b

A = l × (10 - l)

A = 10l - l²

So now our area function has only 1 kind of variable in it, that is l for the length.

So the function of length for a rectangle with perimeter of 20 m is,

A = 10l - l²

Note:

The function (A = 10l - l²) only works if the length is less than or equal to 10. Otherwise the result will be negative, and you don't want a rectangle with a negative area right?