College Algebra, Chapter 1, 1.3, Section 1.3, Problem 84

Find the length $x$ if the shaded area is $160 in^2$







Let us divide the entire region into two parts let it be $A_1$ and $A_2$.







So, $A_1 = 14x$ and $A_2 = (13 + x)(x)$

Thus, the total area $A_T$ is equal to the sum of $A_1$ and $A_2$.


$
\begin{equation}
\begin{aligned}

A_T =& 14x + (13 + x)(x)
&& \text{Model}
\\
\\
160 =& 14x + 13x + x^2
&& \text{Substitute the given adn apply distributive property in the right side of the equation}
\\
\\
160 =& 27x + x^2
&& \text{Combine like terms}
\\
\\
x^2 + 27x - 160 =& 0
&& \text{Subtract 160}
\\
\\
(x + 32)(x - 5) =& 0
&& \text{Factor}
\\
\\
x + 32 =& 0 \text{ and } x - 5 = 0
&& \text{ZPP}
\\
\\
x =& -32 \text{ and } x = 5
&& \text{Solve for } x
\\
\\
x =& 5 in
&& \text{Choose } x > 0


\end{aligned}
\end{equation}
$