Find a formula for the described function.

Question

A rectangle has area 9 m². Express the perimeter of the rectangle as a function of the length L of one of its sides.

P(L) =  

 

Answer 1

The formula for the area of a rectangle is A = LW.

The formula for the perimeter is P=2L + 2W.

The goal is to have P depend on L only.

So solve the first formula (given that A = 9) for W in terms of L.

LW = 9

W= 9/L

Now substitute that expression for W in the perimeter formula:

P = 2L + 2(9/L) = 2L + 18/L.

If you want, get a common denominator and combine to one fraction:

P = (2L² +18)/L

Finally you can use function notation:

P(L) = (2L²+18)/L

If the length is 4 (for example):

P(4) = (2(16) + 18)/4 = 25/2.

Note that when L = 4, P = 9/4 and A = 4(9/4) = 9.

Answer 2

A = LW = 9

W=9/L

P= 2(L+W)= 2(L+ 9/L)

P(L) = 2L + 18/L