We know that
• The length is 7 meters more than double the width.
,• The area is 99 square meters.
First, we express the relationship between the length and the width.
[tex]l=7+2w[/tex]We know that the area is
[tex]A=l\cdot w[/tex]Replacing the first expression, and the area.
[tex]99=(7+2w)\cdot w[/tex]Let's solve for w.
[tex]\begin{gathered} 99=7w+2w^2 \\ 2w^2+7w-99=0 \end{gathered}[/tex]Now, we use the quadratic formula.
[tex]w_{1,2}=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]Where a = 2, b = 7, and c = -99. Let's replace all these values in the formula.
[tex]\begin{gathered} w_{1,2}=\frac{-7\pm\sqrt[]{7^2-4\cdot2\cdot(-99)}}{2(2)}=\frac{-7\pm\sqrt[]{49+792}}{4} \\ w_{1,2}=\frac{-7\pm\sqrt[]{841}}{4}=\frac{-7\pm29}{4} \end{gathered}[/tex]Now, we rewrite it into two equations.
[tex]\begin{gathered} w_1=\frac{-7+29}{4}=\frac{22}{4}=5.5 \\ w_2=\frac{-7-29}{4}=-\frac{36}{4}=-9 \end{gathered}[/tex]The width is 5.5 meters because it can't be represented by a negative number.
We use the width to find the length.
[tex]\begin{gathered} l=7+2w \\ l=7+2(5.5) \\ l=7+11=18 \end{gathered}[/tex]The length is 18 meters.
