This one is tricky. Let's assume that the heading is 105 degrees clockwise from true north. Then the driver's
course is 15 degrees below the x-axis. We know that the slope of a line is m=tan(angle).
In this case m = tan(-15) = -0.2679, so we can write the equation of the line as y = -0.2679x
Since radio puts out a signal with radius 25 miles, the signal is a circle with radius 25.
We know that the radio is 30 miles east of the driver. So we can write the equation of the signal as
(x-30)2 + y2 = 252 or (x-30)2 + y2 = 625.
So the radio station will be in range between the two intersection points between the lines y=-0.2679x and the circle (x-30)2 + y2 = 625. To solve for the intersection points, solve (x-30)2+(-0.2679x)2 = 625 for x.
The two points of intersection are (5.03, -1.347) and (50.94, -13.64)
Determine the distance from point of intersection to the origin. d1=sqrt(5.032+(-1.347)2) = 5.207
So total distance where the driver is in range is 52.73-5.207 = 47.5 miles. I'll let you figure out the other part.