Pre-calc radio station range type problem. Please help!!

Question

You are driving on a course of 105 degrees. A radio station that is due East and 30 miles away from your current location puts out a signal with a 25 mile radius. If you continue on your 105 degree course, in how many miles will the radio station be in range? Once you are in range, how many more miles until you are out of range?

Answer 1

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

d₂=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.

Answer 2

The problem makes sense only if the car is on a "bearding" of 105°

The route of the car is y = -0.2679x (tan 15°)

The range of the signal is x² + (y - 30)² = 625

Determine the two points of intersection.

Determine the distance between the two points.