Christopher J. answered • 05/12/20

Berkeley Grad Math Tutor (algebra to calculus)

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 }+ y^{2 }= 25^{2} or (x-30)^{2} + y^{2} = 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 }+ y^{2 }= 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. d_{1}=sqrt(5.03^{2}+(-1.347)^{2}) = 5.207

d_{2}=sqrt((50.94)^{2}+(-13.64)^{2})=52.73

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.