Radio direction finders are placed at points A and B, which ar...

Question

Radio direction finders are placed at points A and B, which are 3.46 mi apart on an east-west line, with A west of B. From A the bearing of a certain radio transmitter is 47.7°, and from B the bearing is 302.5°. Find the distance of the transmitter from A.

Accepted Answer

Hello, today we are going to be solving the following word problem using methods of trigonometry. So we are told that we have two telecommunication towers, K and L which are situated on a north south line with K located to the north of L. The distance between the telecommunication towers is 8.34 miles from K. The bearing of a power transmission tower is 126.3 degrees. And from L, the bearing is 73.8 degrees. We want to determine the distance between K and the power transmission tower. In order to solve this problem, let's go ahead and draw an image. So we know that we are working with two towers, K and L and they are located on a north south line with K located to the north of L. We can represent this by drawing a vertical line with two points at the end of this line. Since K is located to the north of L, we will position K at the top of this line and L at the bottom of this line, we are told that the distance between the two towers is 8.34 miles. So we know that the length of this line is 8.34 miles. We also know that we are looking at another power transmission tower at some angle, we can represent this tower as the point C and we are looking at this tower at two angles at K and L. Now we are told that K has the bearing of the transmission tower. The angle of this is 126.3 degrees. So we can represent this bearing as 126.3 degrees. We are also told that the bearing from tower L to the transmission tower is 73.8 degrees. So we can represent the angle L at 73.8 degrees. And our goal is to determine the distance between K and the transmission tower, we will go ahead and label the side as little L. So how can we approach this problem? Well, we're representing all the information as a triangle. So before we continue with this problem, let's go ahead and solve for all three sides of the G of the triangle that we drew starting with the angle K, we know that the bearing at angle K is 126.3 degrees. But we want the angle that is located inside of the triangle. We have extended the length of this, of the line to go past K and we are representing this line as a vertical line recall that the total degrees for any straight line is 180 degrees. And we know that the bearing is 100 and 26.3 degrees. So if we were to subtract 126.3 from 180 that will give us the angle located inside the triangle. So we are taking 180 degrees and subtracting 126.3 degrees. That will give us a value of 53.7 degrees for angle K. So let's go ahead and label that. Now we need to solve for angle C, we can use the, the some property for angles of triangle. Now, the property states that the sum of all three angles within a triangle must equal to 180 degrees. So in order to solver C, we can take angle L which is 73.8 degrees at angle K which is 53.7 degrees and at the unknown angle C and this should give us an output of 180 degrees. And we can use this equation to sulfur angle C 73.8 plus 53.7 will give us the value of 127.5 degrees. Now, what we are going to do is we are going to subtract 127.5 to both sides of the equation. This will give us the value of angle C is equal to 52.5 degrees. So we will label that on the triangle. Now, now we have all three angles within the triangle. How can we use this information to solve for little L the side length? What we can do is we can use the law of signs. Now, the law of signs states that we have three equivalent ratios, we can set the sine of angle L divided by little L is equal to sine of angle C divided by little C is equal to sine of angle K divided by little K. Now, in order to solve for the angle L, we can ta or for the side length L, we can take any two of the ratios and set them equal to each other. Looking at the picture, we have the side length resec respective to angle C and we have the angle C, we have the angle L but we are solving for the side length L. So what we can do is we can set the ratio with respect to L equal to the ratio with respect to C. And use that to solve for little L which is the side length we are looking for. So using the law of signs, we can set sine of the angle L divided by the side length L equal to sine of angle C divided by the side length C before we plug in any values. Let's go ahead and cross multiply the two values. When cross multiplying, we will have the sink L multiplied by the sine of angle C equal to the side length C multiplied by sine of the angle L. In order to solve for the side length L, we will divide both sides of this equation by sine of C. And that will leave us with the side length L is equal to the side length C multiplied by sine of angle L divided by sine of angle C. Now we will go ahead and plug in the information we have. We know that the length of sightsee is 8.34 miles. We are going to multiply this value by sine of angle L which is 73.8 degrees and divide this value by sign of angle C which we solved to be 52.5 degrees. At this point. I would recommend using a calculator and plugging in this value into a calculator. When doing so and rounding to one decimal place, you will get the side length L is equal to 10.1 miles. This is going to be the distance between the communication tower K and the transmission tower C. And with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Radio direction finders are placed at points A and B, which are 3.46 mi apart on an east-west line, with A west of B. From A the bearing of a certain radio transmitter is 47.7°, and from B the bearing is 302.5°. Find the distance of the transmitter from A.

Key Concepts

Bearings and Angle Measurement

Triangle Formation and Law of Sines

Coordinate Geometry and Vector Representation

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