Chapter 1.2, Problem 74E

To determine

To Find: The area $A$ as a function of $x$ and determine the domain.

Answer to Problem 74E

  $A=\frac{x^2}{2x-4}$

Domain: $x>2$

Explanation of Solution

Given: A right triangle is formed in the first quadrant by the $x$ axis and $y$ axes and a line through the point $(2,1)$ .

  

Area of triangle

  $\begin{array}{l}A=\frac{1}{2}base(height)\\ A=\frac{1}{2}xy\end{array}$

Now

The points $(0,y),(2,1),(x,0)$ all lie on the same line

Thus

The slopes between any pair of points are equal.

i.e.

  $\begin{array}{l}\frac{1-y}{2-0}=\frac{1-0}{2-x}\\ 1-y=\frac{2}{2-x}\\ y=\frac{x}{x-2}\end{array}$

Substitute $y=\frac{x}{x-2}$ in the area of the triangle

We get

  $\begin{array}{l}A=\frac{1}{2}xy\\ A=\frac{1}{2}x\left(\frac{x}{x-2}\right)\\ A=\frac{x^2}{2x-4}\end{array}$

Hence, the area is $A=\frac{x^2}{2x-4}$

And

Domain: $x>2$