Question

If the sum of base and height of a right triangle is $20cm$, what is the maximum area of the triangle?

Answer 1

Step 1. Find the maximum area of the triangle.

Given, The sum of the base and height of a right triangle is $20\mathrm{cm}$.

Consider that the base of the triangle is $x\mathrm{cm}$ and the height of the triangle is $(20-x)\mathrm{cm}$.
The formula of the area of the triangle is $A=\frac{1}{2}·\mathrm{base}·\mathrm{height}$. Where, $A$ is the area. Then,
$\begin{array}{rcl}A& =& \frac{1}{2}\times x\times (20-x)\\ & =& 10x-\frac{1}{2}{x}^2\end{array}$
Step 2. Simplify the above expression with respect to $\mathit{x}$.

Let us derive the expression $A=10x-\frac{1}{2}{x}^2$ as follows:
$\frac{dA}{dx}=10-x$
$\begin{array}{rcl}\frac{dA}{dx}& =& 0\\ & \Rightarrow & 10-x=0\\ & \Rightarrow & x=10\end{array}$
The maximum value occurs when $x=10$. Then,
$\begin{array}{rcl}A& =& \frac{1}{2}\times 10\times 10\\ & =& 50{\mathrm{cm}}^2\end{array}$

Hence, the maximum area is $\mathbf{50}{\mathbf{}\mathbf{cm}}^{\mathbf{2}}$.