To find: The function that models the area of rectangle in terms of length of one of its sides. The perimeter of rectangle is 20 ft.

Question
Modeling data distributions
asked 2021-01-16
To find: The function that models the area of rectangle in terms of length of one of its sides. The perimeter of rectangle is 20 ft.

Answers (1)

2021-01-17
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 \(\displaystyle{A}={10}{x}\ -\ {x}^{{{2}}}\).
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Suppose the manufacturer of widgets has developed the following table showing the highest price p, in dollars, of a widget at which N widgets can be sold.
\(\begin{array}{|c|c|} \hline Number\ N & Price\ p\\ \hline 200 & 53.00\\ \hline 250 & 52.50\\\hline 300 & 52.00\\ \hline 350 & 51.50\\ \hline \end{array}\)
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