Question

# This homework question has to do with modeling with functions.

Modeling
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.

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
$$\displaystyle\Rightarrow{2}{\left({x}+{b}\right)}={80}$$
$$\displaystyle{2}{x}+{2}{b}={80}$$
$$\displaystyle{2}{b}={80}-{2}{x}$$
$$\displaystyle{b}={\frac{{{80}-{2}{x}}}{{{2}}}}$$ (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 $$\displaystyle{A}={40}{x}-{20}{x}^{{{2}}}$$