Chapter 8.5, Problem 33E

To determine

To find: the area of the region.

Answer to Problem 33E

The area of the triangle region is 250 sq. miles.

Explanation of Solution

Given information:

A large region of forest is infested with gypsy moths. The region is triangular, as shown in the figure. From vertex $A$ , the distances to the other vertices are 25 miles south and 10 miles east, and 20 miles south and 28 miles east.

Calculation:

Now let us draw the axes in above figure as shown below

Here we can see that the coordinate of $A,B$ , and $C$ are $A\left(0,25\right),B\left(10,0\right)$ and $C\left(28,5\right)$ . The area of a triangle whose vertices are given as $\left(x_1,y_1\right),\left(x_2,y_2\right),\left(x_3,y_3\right)$ can be found out using the formula:

  $\pm \frac{1}{2}\left|\begin{array}{ccc}x& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|$

Where the symbol $\pm$ indicate the appropriate sign should be chosen to yield a positive area.

By using graphing utility we can find the determinant as

  $\begin{array}{c}\pm \frac{1}{2}\left|\begin{array}{ccc}x& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=\frac{1}{2}\left|\begin{array}{ccc}10& 0& 1\\ 28& 5& 1\\ 0& 25& 1\end{array}\right|\\ =\frac{1}{2}\left(500\right)\\ =250\end{array}$

Hence the area of the triangle region is 250 sq. miles.