Question

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

Short answer

The expression for the area of the rectangle as a function of length is \({\bf{A}}\left( {\bf{L}} \right){\bf{ = 10L - }}{{\bf{L}}^{\bf{2}}}\)and the domain of the function is\(\left( {{\bf{0,10}}} \right)\).

Step-by-step solution

Determine the equation for the perimeter of the rectangle

The perimeter of a rectangle is given as:

\(p = 2\left( {L + B} \right)\)

Here,\(L\)and\(B\)are the length and breadth of the rectangle,\(p\)is the perimeter of the rectangle.

Substitute all the values in the above equation.

\(\begin{array}{c}20\;{\rm{m}} = 2\left( {L + B} \right)\\L + B = 10\\B = 10 - L\end{array}\)

Determine the expression for the area of the rectangle as a function of length

\(A\left( L \right) = L \cdot B\)

Substitute all the values in the above equation.

\(\begin{array}{l}A\left( L \right) = L\left( {10 - L} \right)\\A\left( L \right) = 10L - {L^2}\end{array}\)

The area of a rectangle should be greater than zero.

\(\begin{array}{c}10L - {L^2} > 0\\L\left( {10 - L} \right) > 0\end{array}\)

Therefore,\(L \in \left( {0,10} \right)\)

Thus, the expression for the area of the rectangle as a function of length is \(A\left( L \right) = 10L - {L^2}\) and the domain of the function is\(\left( {0,10} \right)\).