Radio Coverage

Part of the Triangles Without Right Angles section of Coddy's Trigonometry journey. Lesson 7 of 37.

A road leaves town A at 25° to the line from A to a radio mast M. The mast is 16 km away. The signal reaches 10 km. How many times does a car on the road cross the edge of the signal?

$A road from A with two points P₁ and P₂ at the same distance 10 from the mast M$

The road comes closest to the mast at a distance $h = 16\sin\left(25^{\circ}\right)$ , about 6.76 km. 10 km is between 6.76 and 16, so the car crosses the edge twice. That means two triangles. Signal less than h: none. Exactly h: one, the road only touches the edge. Between h and AM: two. AM or more: one.

Challenge

A road leaves town A at 30° to the line from A to a radio mast M. The mast is 18 km away. The signal reaches 15 km. A car enters the signal at P₁ and leaves it at P₂. D is the point of the road closest to the mast. Find the distance d from D to P₂. Done means d is a number of kilometres.

Then press Check.

Try it yourself

$Use the figure to find d$