## Precalculus (6th Edition) Blitzer

The maximum area of the triangle is $100$ square inches.
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.