Respuesta :
Given that:
- The rug should be no smaller than 48 square feet and no bigger than 80 square feet.
- The length is 2 feet more than the width.
The formula for calculating the area of a rectangle is:
[tex]A=lw[/tex]Where "l" is the length and "w" is the width.
In this case, you know that:
[tex]l=w+2[/tex]Therefore, you can express the formula for the area of the rectangular rug as:
[tex]A=(w+2)w[/tex]Now you can set up the following inequality:
[tex]48\leq(w+2)w\leq80[/tex]Solve for "w", in order to find the range of possible values for the width:
1. Apply the Distributive Property:
[tex]48\leq(w)(w)+(2)(w)\leq80[/tex][tex]48\leq w^2+2w\leq80[/tex]2. Set up the first inequality:
[tex]48\leq w^2+2w\text{ }[/tex]Subtract 48 from both sides:
[tex]48-48\leq w^2+2w-48[/tex][tex]0\leq w^2+2w-48\text{ }[/tex]Factor it and solve for "w":
[tex]\begin{gathered} 0\leq(w-6)(w+8) \\ \\ 6\leq w \\ -8\leq w \end{gathered}[/tex]3. Set up the second inequality and apply the same procedure:
[tex]w^2+2w\leq80[/tex][tex]\begin{gathered} w^2+2w-80\leq0 \\ (w-8)(w+10)\leq0 \\ w\leq8 \\ w\leq-10 \end{gathered}[/tex]The width must be positive.
Hence, the answer is:
[tex]6\leq w\leq8[/tex]