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You find very tall tree one day and decide to measure the height of the tree using pendulum You tie small rock to rope that is also tied to branch at the top of the...

Question

You find very tall tree one day and decide to measure the height of the tree using pendulum You tie small rock to rope that is also tied to branch at the top of the tree So thal When the rope is vertical the rock is very close to the ground. You then release the rock from rest at an initial angle of 59. You measure the velocity of the rock when it returns t0 equilibrium to be 1,5 mls_ How tall is the tree?

You find very tall tree one day and decide to measure the height of the tree using pendulum You tie small rock to rope that is also tied to branch at the top of the tree So thal When the rope is vertical the rock is very close to the ground. You then release the rock from rest at an initial angle of 59. You measure the velocity of the rock when it returns t0 equilibrium to be 1,5 mls_ How tall is the tree?



Answers

Two people decide to find the height of a tree. They position themselves 35 feet apart in line with, and on the same side of, the tree. If they find the angles of elevation from the ground where they are standing to the top of the tree are $65^{\circ}$ and $44^{\circ},$ how tall is the tree?

So let's start the problem. There's a guy who is standing 30 ft. I'll show you the scenario. Okay this Will be the three. Okay. Like this, it would be a pretty, this is the top, this is the base and there's a guy standing okay like this who's standing. So the angle of elevation will be from his eyes over here to this point. Okay And angle of elevation is uh to the top is 50. And angle of depression. Okay? Uh to the down is 10. Okay. Uh yeah to the base of the tree. So this is the scenario and this much distance. It has given us 30 30 ft. Okay so I just rather triangles. Okay so that you can understand it better. So this will be the two right angle triangles. Okay so in this case uh here the angle is just once again you are the angles 50 degree. Here the angle will be 10 degrees. This is the angle of depression. Okay and uh This much the distance. Okay this is 13 till the spaces 30. And this is three, it is the top and this is the best. Okay so I just name like T. B. And this will be a P. Thank you. Okay. T. V. P. Q. Okay so let's just write the questions. We actually have to find their daughter length. Okay, the height of the great. Okay so this one, I will assume it's us X. And this as white. Okay, this much land so that it becomes easy for me. Okay so I'll just uh this becomes quite confusing. I'll just label it using different colors so you can understand. Okay uh This is T. This is Q. This was P. This is base be. Okay so let's just start. Okay so I like in triangle T. P. Q. Is right angle over here. Okay so what I can like this uh santa. Okay you know what I stand data that is perpendicular upon base. That is opposite side up, divide by and just inside. So I'll be writing state. Okay or as I like like the stand 50 will be equal to TP upon how much speed you. Okay DPM Bond. Thank you. So what is the value for T. P. Uh D. P. Is actually X. And P. Two is 30. Okay so X. Will be 10 50 in 2 30. Okay so let me just find what stand 30 and at 10 30 it's 10 50. Uh 10 50 is actually 1.1917 into 30. Okay 1.197. And then I use the calculator just one second. 1.1917 into 30. Okay this terms us 35.751. So we have got the value of excess 35.75 ft. Okay now we will use the different the lower triangle. So the triangle will be B. P. Q. Okay right angle at B. B. P. Q. Okay so you just have to look from here. Okay from this point you have to look so in triangle be PQ. If I take a 10 10. Okay 10 10 it will be actually what this upon this particular upon base. So this will be why upon 30. Okay and let me just find what is 10 10? Okay. 10 10 is 0.16. Okay and just write this so that you can understand BP upon PQ again. So 10 10 is 0.1763. It is a culture by by 30. So I will be 30 in 20.1763. So I'll just use a calculator. Mm Yeah 30 into 0.1763. This comes as a 5.28. Okay bicycle to 5.28 uh 9 ft. Okay why is this much? So I'll just take the figure to the next page to tell you how much will be the height. Okay copy I did I remove it? Okay so I actually moved it. I'm sorry King. Oh yeah and just take it on to the next page. Okay copy. Here we go. Okay We have got Y. is equal to 5.289. I just found it a fast 5.29. Okay and the other one X. is equal to 30. 5.75. Okay it's okay so uh we told that higher after three will be X. Plus Y. C. This much and plus this much. If you can see visualize this was a. Three. So it was forming an angle of elevation at this point. So this much length. We found us eggs. And this much is why. Okay so the height of the play will be X plus Y. So this will be uh In just one second, X. is actually 35.75. Okay. Yeah. 35.75 plus 5.29. Okay. Father, don't get up to 5.34 years. The calculation. Okay so five point 30, this comes as five. This is 10 1 goes over there. Uh You do miscarry rounding off this giving us the different answer. Okay so I didn't do nothing itself. So 5.29 35 points and five and 5.295 plus nine is a homage. 14 14 plus one. This 11 1 goes over there, five plus five is 10. And this is 41.14. Okay feet So height up the tree is 41.14. If we round of it is 41. Okay, so you're like hired up

When we are doing trigonometry application questions, it does help to draw out the situation and see what trying over dealing with. So we know we have a tree and you're looking for its height. So let's call the trees height H. Now we know that there's a point 105 feet from the tree, and from there the angle between the horizon and the direction to look at the top of the tree. That is to say, this angle is 65.3 degrees. Now notice we have a right triangle, since this the dotted lines horizontal on the tree is vertical. So ah, we are going to solve for H. And to do that, it's best to find a trig ratio of the known angle that consists of the unknown H and the known side. 105 feet so we can see in the Figure H is opposite to the known angle, and the 105 feet decided is adjacent to that angle, a trick function that uses opposite energies into his tangent. You can see the tensions of that angle 65 0.3 degrees. That is gonna be to equal to H the opposite Over 105. The height. Sorry. The be adjacent. Now, um, we just flip the sides and multiply both sides by 105 and we get at age is equal to 105 times the tangent of 65.3 degrees. And we enter the sense of the calculator to get that, um, h is equal to 228 feet. And I specifically, uh, kept three significant figures because the length that was given in the question 105 was, uh, ridden with three significant figures, So our final answer is 228 feet.

In this problem, we have toe find what is the height off the tree If the angle evaluation from the ground at each person pollution to the top of the tree is 40 year daily now, first will make a diagram now in the diagram. This is best. This is not. This is south. This is east and the points are B C A and D now let 83. Since both persons are seeing distance from the three. Therefore, let distance off A and B, B X and A and C off X on the distance between B and C is given as 35 so we have leg. A B is equals toe A C is equals toe eggs and indeed is equals toe. Three. So in Triangle A B C in the square plus ISI square is equals toe PC square, so X square plus X square is equal to 25 square. We get X squared equals toe one toe fight over to get exit. It was toe under rule. One toe fight over to that is, except it was 2 20 4.7. Moving now entangled D A. C Angle D. A. C is equals toe 48 degrees and engine 48 is equals toe really over a c we get 80 is equals toe the A C multiply 10 jin 48. Therefore, we got 24.74 25 1.11 we get 80 is equals to 27.47 ft. Therefore the height is 24.47 ft for the final solution is this one?

Question. We can draw the picture here, So wait on. We have the Mr Hayes 18. This one is 62. Anderson is gentle. Tree hair is 75. Therefore, let me conduct I b and C and the anger be totally We go to the 62 plus gentle tree, and we go ju 85. Therefore, the angles say we go to 180 minus 85 minutes. 75 a code. You won 16 so you could joke 20 degree here and now we can find ah ah, using the law off sign. Then we get going to find, uh, uh be small. Be here. So be off a sigh off 85 were ego do I d all of us. I'm the angle. See gente their found obey weak. Oh, you're a three time Cy off. Ah, 85 dividing us a mature lt never a culture. I date him Sy under 85 then the wedding by silent drunk a Got Nicole to 233. So this is how it a go, Jude. 233 feet


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