34.26miles from 49.58miles of the drive (69.11% of the drive).
A problem of Analytic Geometry. This question can be solved using the resulting values from equaling the equations of the line (for the drive) and circle (with radius equals 30miles for radio transmitter broadcasting), and calculating the total distances from coincident points between circle and line, and total drive.
A graph is attached showing part of the circle and line with coincident points.
Assuming transmitter is located (0, 0). From a city 33 miles north of the transmitter (0, 33) to a second city 37 miles east of the transmitter (37, 0).
Then, the equation for the line is (taking the two points from the start to the end):
The equation for the circle whose center (0, 0) is:
Equaling  and , to determine the points where starting to receive the radio transmitter signals to the point where finishing those signals:
Solving this equation for <em>x</em>, we have two solutions for it (from <em>WolframAlpha</em>):
Then, using , the corresponding values for <em>y</em> are:
Well, the distance of the drive is:
The total distance traveled is:
Thus, during the drive, the signal of the radio transmitter was picked up for 34.26miles from the total of the 49.58 traveled, that is, a fraction equivalent to or 69.11% of the drive.