Solve each problem. See Examples 1–4. Distance across a Lake To ...

Question

Solve each problem. See Examples 1–4.  Distance across a Lake To find the distance RS across a lake, a surveyor lays off length RT = 53.1 m, so that angle T = 32°10' and angle S = 57°50'. Find length RS.

Accepted Answer

Welcome back everyone in this problem to find the distance across the pond A B on our diagram, an engineer uses field tape to measure AC and obtain the length of 31.2 m. This length results in an angle measure of C equals 41 degrees, 12 minutes and B equals 48 degrees, 48 minutes. And we want to find the length A B rounded to one decimal place for our answer choices. A is 24.6. B is 19.3. C is 31.3 and D is 27. and all are in meters. Now, what do we know here? Well, in order to solve any problems to figure out the length of A B as indicated on the diagram, we need to know what kind of triangle we're working with. What do we already know? Well, we know that the angle at C is 41 degrees and 12 minutes. And we know the angle at B is, let me put it here 48 degrees and 48 minutes. So if we can figure out our unknown angle at A, then we'll be in A better position to. So for the length of A B now recall that the sum of interior angles, some of interior angus off a triangle is 180 degrees. So since we know that, that means that A plus B plus C should be equal to 180 degrees, so if we're going to solve for A, that means we're going to subtract B and C from degrees. So that would be 180 minus 48 degrees. 48 minutes. That's B plus 41 degrees, 12 minutes. That's angle C now 48 degrees plus 41 degrees equals 89 degrees and 48 minutes plus 12 minutes equals one degree. So our sum would be 1 80 minus that means A equals 90 degrees. And we can show that on our diagram. No, that's a very important result because that tells us then that the triangle we're working with is a right triangle and it's a right triangle with all right angle at a, what else do we know on our diagram? We also know that the length of AC is 31.2 m. OK. So let me put it on our diagram here, 31.2 m. So if we are solving for the length A B in a right triangle, then we can try to use one of the trigonometric ratios because we know the angles in our right triangle. So taking the angle 41 degrees and minutes. OK. Angle C for angle cab, you know what, let me redraw it with a diagram down here. So on our diagram, OK. We know that uh we have ABC and at sea, our angle is 41 degrees 12 minutes. We already know that AC is 31.2 m and we're trying to figure out the length of A B. So in relation to our anger, we know that side A B is the opposite side. And we know that side AC is the adjacent side. What trigonometric ratio links the opposite and adjacent sides? Well, it would be the tangent because recall that the tangent of an angle is the ratio of the opposite side to the adjacent side. So that tells us then that the tangent of our angle 41 degrees 12 minutes equals the opposite side, which is A B divided by the adjacent side, which is 31.2. If we multiply both sides by 31.2 it means then that A B equals 31.2 multiplied by the tangent of 42 degrees 12 minutes. When we calculate that value, then we should get A B to be equal to 27.31353528. And it goes on depending on the calculator you're using. But record, we only want our answer to the R to one decimal place. So to one decimal place A B would be equal to 27.3 m. When we look back on our answer choices, that would have been answer choice. D thanks a lot for watching everyone. I hope this video helps.

Solve each problem. See Examples 1–4. Distance across a Lake To find the distance RS across a lake, a surveyor lays off length RT = 53.1 m, so that angle T = 32°10' and angle S = 57°50'. Find length RS.

Key Concepts

Law of Sines

Angle Measurement

Triangle Properties

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