Concept used:
Area of rectangle is defined as,
\(\displaystyle{A}={l}\ \cdot\ {b}\)
Here, A is area, l is length and b is width.
Calculation:
Consider the length of one of its sides of rectangle as x.
The perimeter of a rectangle is defined as,
\(\displaystyle{P}={2}\ {\left({l}\ +\ {b}\right)}\)
Here, P is perimeter.
The perimeter of rectangle is given as 20 ft.
Substitute 20 for P and x for l in above equation to the value of b.
\(\displaystyle{20}={2}\ {\left({x}\ +\ {b}\right)}\)
\(\displaystyle{\frac{{{20}}}{{{2}}}}={x}\ +\ {b}\)
\(\displaystyle{10}={x}\ +\ {b}\)
\(\displaystyle{b}={10}\ -\ {x}\) Therefore, the value of b is \(\displaystyle{\left({10}\ -\ {x}\right)}\) Substitute x for l and \(\displaystyle{10}\ -\ {x}\) for b in equation to obtain the model that express area of the rectangle \(\displaystyle{A}={x}\ {\left({10}\ -\ {x}\right)}\)
\(\displaystyle={10}{x}\ -\ {x}^{{{2}}}\) Here, x is always greater than zero and less than 10 to define the area therefore the value of x is always lies in between 0 and 10. Conclusion: Thus, the function that models the are of rectangle in terms of length of one of its sides is \(\displaystyle{A}={10}{x}\ -\ {x}^{{{2}}}\).
\(\displaystyle{\frac{{{20}}}{{{2}}}}={x}\ +\ {b}\)
\(\displaystyle{10}={x}\ +\ {b}\)
\(\displaystyle{b}={10}\ -\ {x}\) Therefore, the value of b is \(\displaystyle{\left({10}\ -\ {x}\right)}\) Substitute x for l and \(\displaystyle{10}\ -\ {x}\) for b in equation to obtain the model that express area of the rectangle \(\displaystyle{A}={x}\ {\left({10}\ -\ {x}\right)}\)
\(\displaystyle={10}{x}\ -\ {x}^{{{2}}}\) Here, x is always greater than zero and less than 10 to define the area therefore the value of x is always lies in between 0 and 10. Conclusion: Thus, the function that models the are of rectangle in terms of length of one of its sides is \(\displaystyle{A}={10}{x}\ -\ {x}^{{{2}}}\).
