Question

This homework question has to do with modeling with functions.

Modeling
ANSWERED
asked 2021-08-18
This homework question has to do with modeling with functions.
1.) A rectangle has a perimeter of 80 ft. Find a function that models its area A in terms of the length x of one of its sides.
Can you please explain exactly what you did and what you used to find the formula.

Expert Answers (1)

2021-08-19

Step 1
Perimeter of rectangle \(\displaystyle={80}{f}{t}\)
let length of rectangle be x ft
Perimeter of rectangle formula \(\displaystyle={2}\) (length+breadth)
where x is length
b is breadth
\(\displaystyle\Rightarrow{2}{\left({x}+{b}\right)}={80}\)
\(\displaystyle{2}{x}+{2}{b}={80}\)
\(\displaystyle{2}{b}={80}-{2}{x}\)
\(\displaystyle{b}={\frac{{{80}-{2}{x}}}{{{2}}}}\) (1)
Step 2
Area of Rectange formula \(\displaystyle=\text{length}\times\text{breadth}\)
\(\displaystyle\Rightarrow{A}={x}\times{b}\) (2)
Where \(\displaystyle{A}\rightarrow\) Area
\(\displaystyle{x}\rightarrow\) length
\(\displaystyle{b}\rightarrow\) breadth
Substitute b value in terms of x from 1 in 2
\(\displaystyle\Rightarrow{A}={x}\times{\left({\frac{{{80}-{2}{x}}}{{{2}}}}\right)}\)
Step 3
\(\displaystyle{A}={\frac{{{80}{x}-{4}{x}^{{{2}}}}}{{{2}}}}\)
\(=\frac{\not{2}(40x-2x^{2})}{\not{2}}\)
\(\displaystyle={40}{x}-{2}{x}^{{{2}}}\)
The function that models area is given by \(\displaystyle{A}={40}{x}-{20}{x}^{{{2}}}\)

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