Chapter 8.3, Problem 63PS

Solution

To find: if the location 40 miles north and 25 miles west of the bridge located within the bats feeding region.

Yes, the location is in the feeding region.

Given information:

The bats have an estimated feeding range of 50 miles.

Formula Used:

The Distance Formula

The distance d between $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is:

  $d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Standard equation of a circle with Center at the Origin

The standard form of the equation of a circle with center at $\left(0,0\right)$ and radius r is as follows:

  $x^2+y^2=r^2$

Explanation:

Let the position of the bridge is at Origin $\left(0,0\right)$ . Then, the location 40 miles north and 25 miles west is written as coordinates as $\left(25,40\right)$

So, the distance between the tower and the person is:

  $\begin{array}{c}d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(25-0\right)}^2+{\left(40-0\right)}^2}\\ =\sqrt{625+1600}\\ =\sqrt{2225}\end{array}$

Now, it is given that the bats have an estimated feeding range of 50 miles. So, the region of bats feeding is the circular area around the bridge with 50 miles as radius.

  $\begin{array}{c}{x}^2+y^2={50}^2\\ x^2+y^2=2500\end{array}$

Now, as $2225<2500$ , so the location is in the bats feeding region.