To find the distance AB across a river, a surveyor laid off a ...

Question

To find the distance AB across a river, a surveyor laid off a distance BC = 354 m on one side of the river. It is found that B = 112° 10' and C = 15° 20'. Find AB. See the figure.

Accepted Answer

Welcome back. Everyone in this problem. A group of students in Bangkok, Thailand is performing an activity in the vicinity of the Chao Fria river. The measurements angle and length they obtained are shown in the figure, determine the distance D that runs across the river. For our answer choices. A says it's 278.6 m. B 324.9 m C 311.8 m and D 297.8 m. No, let's try to represent this diagram on our own. OK. Just that it looks a bit clearer. So we notice here that this activity the students are performing forms are forms a triangle. OK? That represents the distances for our given river. OK? And for the distances here, we know that we have sides defk where we know one of the angles is 39 degrees 20 minutes. And we know another angle in our diagram is 115 degrees, 30 minutes. OK? And we oops my bad. We also know, OK, that the side DF is 280 m. And what we're trying to find is the side D that is the distance that runs across the river. Now let's make a note of all the information we have here. So we know we're working with a triangle, we know two angles and we know the side that is adjacent to both of those angles. Now, if we know two angles in the triangle, then we should be able to find the third missing angle. And by finding that third angle that is angle, then we'll at least have one side that's opposite to a known angle. When we have a side that's opposite to unknown angle, then we know we will be able to use the sign law. Now, before we try to get to the same law, let's first figure out that angle angle def or the angle at point E. Now we know angles in a triangle some to 180 degrees. So since that's true angles in a triangle, some to 180 degrees, then that means angle E is going to be equal to 180 degrees minus the sum of the other two angles that is 39 degrees and 20 minutes plus 115 degrees and 30 minutes. So we can use this to figure out angle E. Now, when we work the salt, you should find that you get angle E to be equal to 25 degrees and 10 minutes. OK. So we know that angle E is 25 degrees and 10 minutes. And now we have a pair of known angle and side. Remember we're trying to find side de and we know the angle that's opposite to de, OK. We know that the 39 degree 20 minute angle is opposite to de. So since we have a known pair of angle and sides, like we said earlier, we are confident we can use the sign law to help us figure it out. Because recall that the same law says, well, that the sine law says that for a triangle, OK? The ratio of the sine of an angle in this case, let's say the sine of D to its opposite side D, OK is the same through the triangle. In other words, the sine of D to its opposite side D is the same as the sine of E to its opposite side E and the sine of F to its opposite side F OK? So in this triangle, if we label side D as F, because it's opposite to angle F and we label side FD as E because it's opposite to an E and then ef as side D, OK? Then we know that the sine of F to F is the same as the sine of E to E. OK. In other words, the sign of angle F which is 39 degrees and 20 minutes two, it's opposite side de is going to be equal to the sign of the sign of angle eve which is 25 degrees and 10 minutes to its opposite side. 218 m. No, we have three, we have one unknown in our equation. So we can solve for our unknown de first of all, we can do some cross multiplication here. So when we do that, we find that d multiplied by the sine of 25 degrees and 10 minutes is equal to 218 m multiplied by the sine of 39 degrees and 20 minutes. Therefore, that means if we divide both sides by the sine of 25 degrees, 10 minutes de is going to be equal to 218 sine 39 degrees, 20 minutes divided by the sine of 25 degrees 10 minutes. And now to figure out the value of de, all we have to do is to put this equation into our calculator or this expression rather into our calculator. Now, when you do that, you should find that you get de to be equal to 324.9247197 m. OK? But remember all of our answers were written to the nearest 10th of a meter. So when we rounded to the nearest 10th of a meter here, nine, we have 9/10 beside it, the digit is less than five. So we're going to keep it, which tells us then that d has to be equal to 324.9 m to the nearest 10th. Thus, the distance D that runs across the river is 324.9 m. If we look back on our answer, choices, B is the correct answer. Thanks for watching everyone. I hope this video helped.

To find the distance AB across a river, a surveyor laid off a distance BC = 354 m on one side of the river. It is found that B = 112° 10' and C = 15° 20'. Find AB. See the figure.

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Key Concepts

Law of Sines

Angle Sum Property

Trigonometric Ratios

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