To determine the distance RS across a deep canyon, Rhonda lays...

Question

To determine the distance RS across a deep canyon, Rhonda lays off a distance TR = 582 yd. She then finds that T = 32° 50' and R = 102° 20'. Find RS. See the figure.

Accepted Answer

Welcome back. Everyone in this problem. The geologists from geo experts co are conducting fieldwork at a valley. Their measurements are shown in our figure. We want to determine the distance tu that is the distance across the river. For our answer choices. A says it's 422.3 m. B 397.6 m C 463.8 m and D 377.5 m. Now, if we're going to figure out the distance, Tu let's try to isolate our, our diagram into our own little triangle here. OK. So notice that, oops, I think this is not trying the best. OK. This looks a bit better. So we have our triangle here. OK. Triangle, Tuv, oops and four triangle T UV. Side TV is five. Let me write it here. 587 ft and are angle at sea. Our angle at V is 26 degrees and 30 minutes. And our angle at T OK is 115 degrees 10 minutes. So in other words, we know the values of those two angles. Let me share it in blue to show that we know it. And now we're trying to find the distance. T OK. Now, let's think a little bit about our or a triangle that's formed here. So notice we're trying to find the side tu, we know the anger that's opposite to tu. We know that that's an V in our triangle, we have two angles and with those two angles, that means we should be able to find our third angle that is angle. You recall that angles in a triangle sum to 180 degrees. So since I goes in a triangle, some to 180 degrees, OK, that means to figure out the value of anger, you, we can subtract the other angles from 180 degrees. In other words, U equals 180 minus the sum of 26 degrees and 30 minutes plus 115 degrees and 10 minutes. Now, when we do that, OK, we should find that we get angry you to be equal to 38 degrees and 20 minutes. So now we know the value of anger. You, what does this mean for us? Well, no, OK. Notice that anger you is opposite to the side TV. So we have a pair of anger and known side. When we have a pair of angle, unknown side, we can use the sign law to help us figure out any other unknown angles or sides in the triangle. So since we have, we want to find Tu and we know the opposite to Tu, which is V, we can use that idea to figure out what the length of Tu is. Because recall that from the sign law, OK. The sign law tells us that the sine of an angle to its opposite side or sorry, the ratio of the sine of an angle to its opposite side in a triangle is the same throughout the triangle. In other words, the ratio of the side, in this case, the side T to the sine of anger T is going to be the same as the ratio of side U to the sine of anger U which is the same as V to the sign of V. Now, in this case, OK? Here, if this side is V, if this angle is V, then the side opposite to V is V, the side opposite to U is U, OK? And the side opposite to angle T is T. Now, since we know the length of side U and U, we can use the sign law to help us figure out the length of side V that is T because applying the sine law, then this means that Tu, OK? To the sine of the opposite to it Z which is 26 degrees and 30 minutes is going to be equal to the side. We know 587 ft, ok? You to the sign of you the sign of anger 38 degrees and 20 minutes that we just found. No, we should be able to use that to solve for the side tu because if we multiply both sides by the sine of 26 degrees and 30 minutes, that means T equals 587 sine 26 degrees, 30 minutes. OK. Divided by, I did buy the sign of 38 degrees and 20 minutes. And now to figure out the value of TU, we just need to plug this into our calculator. So let's put this expression into our calculator. What do you find? Well, you should find that T is equal to 422 point sorry 0.2 8 8 1 7 9 2 m. Remember that all of our answers were written to the nearest 10th of a meter. So if we round this value, OK, to that point, we have two in the space of the 10th, the value beside it is greater than five. So we're going to round up. So Tu is then approximately equal to 422.3 m to the nearest 10th of a meter. Therefore, if we go back to our answer choices, A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

To determine the distance RS across a deep canyon, Rhonda lays off a distance TR = 582 yd. She then finds that T = 32° 50' and R = 102° 20'. Find RS. See the figure.

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Key Concepts

Law of Sines

Angle Measurement and Conversion

Triangle Angle Sum Property

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