Standing on one bank of a river flowing north, Mark notices a ...

Question

Standing on one bank of a river flowing north, Mark notices a tree on the opposite bank at a bearing of 115.45°. Lisa is on the same bank as Mark, but 428.3 m away. She notices that the bearing of the tree is 45.47°. The two banks are parallel. What is the distance across the river?

Accepted Answer

Hello, today we are going to be solving the following word problem using methods of trigonometry. We will also be drawing a picture as we read the word problem. So we are told that Dave and Jane are standing on the same side of the street sidewalk as they wait for their school bus. So we can represent the street sidewalk as a horizontal line. Dave is going to be standing on one side of this line and Jane will be standing on the opposite side. We are told that Dave spots a traffic cone on the opposite side of the sidewalk and observed its bearing at 27.85 degrees. So Dave is looking at a traffic cone on the opposite of the street which you can represent ST as he's observing this cone, he's observing it with a bearing of 27.85 degrees. Now, we are told that Jane is 7.4 m away from Dave and saw the traffic cone at a bearing of 351.46 degrees. So there are two pieces of information that are given to us in this sentence, the distance between Dave and Jane is 7.4 m. And Jane is looking at the same traffic cone with the bearing of 351.46 degrees. We want to calculate the width of the street and we can represent the width of the street as a vertical line between T and the street that Dave and Jane are standing on. So there are two things that are going on. We can represent 7.4 m as the sum of the distance between the width of Dave and the width with respect to Jane. So we can represent the distance between this midpoint and Dave as the variable X. And we can represent the distance between the width and Jane as the variable Y. And we know that the sum of the distances X and Y must equal to 7.4 m. This is going to be important for later because in order to solve for W we are going to use right angle trigonometry. Let's go ahead and solve for the angle theta for Dave. Now, we know that Dave is looking at the traffic cone with a bearing of 27.85 degrees. But we know that this vertical line with respect to the distance of the sidewalk forms a right angle, a right angle is equal to 90 degrees. So in order to solve for the angle D within the triangle, we are going to take 90 degrees and subtract 27.85 degrees from this, this is going to give us a value of 62.15 degrees. This is going to be the angle D. Now using right angle trigonometry, we can set the tangent of angle D is equal to W divided by X. We know that the angle D is 62.15 degrees. So substituting this value into the equation will give us tangent of 62.15 degrees is equal to W divided by X. Now remember how we stated that the distance of X and Y is equal to 7.4. Well, let's go ahead and solve for X in this trigonometric equation. If we multiply both sides by X, we will have X multiplied by the tangent of 62.15 degrees is equal to W. And if we divide both sides by this tangent value, we will have X is equal to W divided by tangent of 62.15 degrees, we have a substitution for X with respect to W. We are going to repeat this process for the angle or for the value of Y. So we know that Jane is looking at the traffic cone with a bearing of 351.46 degrees. We are first going to solve for the bearing angle with respect to T. In order to do this, we are going to take 360 degrees and subtract 351.46. This will give us a value of 8.54 degrees. So we know that the opposite bearing angle is 8.54 degrees. And just like what we did with Dave, we know that we have a right angle with respect to this vertical line. If we do 90 degrees minus 8.54 degrees, the angle inside the triangle will be 81.46 degrees. This is going to be the angle J. We will also be using tangent to solve for the value of wine. So we know that tangent of angle J is equal to W divided by Y angle J is 81.46 degrees. So we have tangent of 81.46 degrees is equal to W divided by Y. Now just like X, we are going to solve for Y in the equation, we will multiply both sides by Y leaving us with Y multiplied by tangent of 81.46 is equal to W and dividing both sides by the tangent value will leave us with Y is equal to W divided by tangent of 81.46. Now what we have is a substitution for why? In terms of W. So earlier, we made the equation X plus Y is equal to 7.4. If we plug in the substitutions for X and Y width, in terms of W, we can solve for the value of W. So plugging in the substitutions will give us W divided by tangent of 62.15 plus W divided by tangent of 81.46 is equal to 7.4. What we can do is we can factor out a W on the left hand side of this equation, this will leave us with W multiplied by the quantity one divided by tangent of 62.15 plus one divided by tangent of 81.46 is equal to 7.4. At this point, I would recommend using a calculator to solve for this quantity. When plugging in this quantity into a calculator, we will be left with W multiplied by 0.6785 is equal to 7.4. We will divide both sides of this equation by the decimal. So we have W is equal to 7.4 divided by 0.6785. And with the help of a calculator, W will approximately equal to 10.9 m. This is going to be the width of the street that Dave and Jane are standing on. And with that being said, the answer to this problem is going to be D. 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.

Standing on one bank of a river flowing north, Mark notices a tree on the opposite bank at a bearing of 115.45°. Lisa is on the same bank as Mark, but 428.3 m away. She notices that the bearing of the tree is 45.47°. The two banks are parallel. What is the distance across the river?

Key Concepts

Bearings and Directional Angles

Trigonometric Applications in Triangles

Properties of Parallel Lines and Transversals

Watch next