Question

A small radio transmitter broadcasts in a 50 mile radius. If you drive along a straight line from a city 59 miles north of the transmitter to a second city 64 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?

Answer 1

Draw a picture. Placing the transmitter at the origin of an x-y coordinate system. The starting point is (0, 59) and the end is at (64, 0).

The driving path is described by y = -59/64 * x + 59.
The distance from the transmitter r = x^2 + y^2.

Substituting for y:
r = x^2 + (-59/64 * x + 59)^2

Solve for x such that r = 50. There will be two solutions. Those will be the points where you start and end receiving the radio signal.