#### Answer

The signal can be received in locations that are east or west of the origin up to a distance of 200 miles away. The signal can be received in regions which are within $45^{\circ}$ of the horizontal axis.
The signal can not be received in any locations north or south of the origin, or regions located within $45^{\circ}$ of the vertical axis.

#### Work Step by Step

$r^2 = 40,000~cos~2\theta$
We can see the graph for the values of $\theta$ such that $0 \leq \theta \leq 360^{\circ}$
Note that the graph only includes points where $cos~2\theta \geq 0$
That is:
$0 \leq \theta \leq 45^{\circ}$
$135 \leq \theta \leq 225^{\circ}$
$315 \leq \theta \leq 360^{\circ}$
When $\theta = 0^{\circ}$, then $r = \sqrt{40,000~cos~0^{\circ}} = 200$
When $\theta = 15^{\circ}$, then $r = \sqrt{40,000~cos~30^{\circ}} = 186$
When $\theta = 30^{\circ}$, then $r = \sqrt{40,000~cos~60^{\circ}} = 141$
When $\theta = 45^{\circ}$, then $r = \sqrt{40,000~cos~90^{\circ}} = 0$
When $\theta = 135^{\circ}$, then $r = \sqrt{40,000~cos~270^{\circ}} = 0$
When $\theta = 150^{\circ}$, then $r = \sqrt{40,000~cos~300^{\circ}} = 141$
When $\theta = 180^{\circ}$, then $r = \sqrt{40,000~cos~360^{\circ}} = 200$
When $\theta = 225^{\circ}$, then $r = \sqrt{40,000~cos~450^{\circ}} = 0$
When $\theta = 315^{\circ}$, then $r = \sqrt{40,000~cos~630^{\circ}} = 0$
When $\theta = 330^{\circ}$, then $r = \sqrt{40,000~cos~660^{\circ}} = 141$
When $\theta = 345^{\circ}$, then $r = \sqrt{40,000~cos~690^{\circ}} = 186$
When $\theta = 360^{\circ}$, then $r = \sqrt{40,000~cos~720^{\circ}} = 200$
We can see this graph below.
Since locations where the signal can be received correspond to the interior of the curve, the signal can be received in locations that are east or west of the origin up to a distance of 200 miles away. Note that the signal can be received in regions which are within $45^{\circ}$ of the horizontal axis.
The signal can not be received in regions outside of the curve, so the signal can not be received in any locations north or south of the origin, or regions located within $45^{\circ}$ of the vertical axis.