Question

65-70 Find a formula for the described function and state its domain.

A rectangle has perimeter 20m. Express the area of rectangle as a function of the length of one of its sides.

Short answer

\(A\left( L \right) = 10L - {L^2}\), and its domain is \(5 < L < 10\).

Step-by-step solution

Find the relation between the length and width of the rectangle

Let the length of the rectangle be L and the width be W.

Theperimeterof the rectangle is determined below:

\(\begin{aligned}2L + 2W &= 20\\L + W &= 10\\W &= 10 - L\end{aligned}\)

Write the function for the area

Theareaof the rectangle is determined below:

\(\begin{aligned}A &= LW\\ &= L\left( {10 - L} \right)\\ &= 10L - {L^2}\end{aligned}\)

As the length of the rectangle is negative, \(0 < L < 10\). With the restriction in W, the length L should be \(5 < L < 10\).

So, the area of the rectangle is a function of \(A\left( L \right) = 10L - {L^2}\) , and its domain is \(5 < L < 10\).