Answer
The maximum area of the triangle is $100$ square inches.
Work Step by Step
Let $x$ be the height of the triangle.
Therefore, the base is $40-2x$.
The area of the triangle is:
$\begin{align}
& \text{=}\frac{\text{1}}{\text{2}}\left( \text{base} \right)\left( \text{height} \right) \\
& =\frac{1}{2}x\left( 40-2x \right) \\
& =-{{x}^{2}}+20x
\end{align}$
Which is a quadratic equation with $a<0$ and thus will have a maximum at $x=-\frac{b}{2a}$:
$\begin{align}
& x=-\frac{20}{2\left( -1 \right)} \\
& x=10
\end{align}$
The height of the triangle is $10$ inches.
Thus, the maximum area of the triangle is:
$\begin{align}
& -{{x}^{2}}+20x=-{{\left( 10 \right)}^{2}}+20\left( 10 \right) \\
& =-100+200 \\
& =100\text{ sq}\text{.inches}
\end{align}$
The maximum area of the triangle is $100$ square inches.
