Two bird watchers are standing at points A and B, ontwo sides of a tree. The distance between the birdwatcher Aand a bird on the tree branch is 52 feet. The angle o...

Each of two observers 400 feet apart measures the angle of elevation to the top of a tree that sits on the straight line between them. These angles are $51^{\circ}$ and $65^{\circ},$ respectively. How tall is the tree? How far is the base of its trunk from each observer?

In this problem were given this day now and we have to find the height off the here. The points are A B C and the and the angle. Here are 10.7 degrees, 86.6 degrees, and the distance between A and B is 24 point 8 ft. Now let d c. B the three. So Triangle ABC is a right angle triangle with angle C A B requests for 90 degrees and a C is equals toe e V. In June, it is 6.6 degrees. You get 24.8 95 16.83 before we get aces equals toe 417.4 ft. Now, in Triangle D A C, we get engine a physicals toe D c over a C. Therefore we get this is equals toe a C multiply engine eight, we get 417.4 multiplied engine 10.7. They get 417.4, maybe 5.188 we get. This is equals to 78.86 ft. Hence the height off the tree is 78.8 16. So the final solution is this one

So we have to observers, uh, and they're birdwatching. They're burning. So we know that our two observers, which is used to put his dots here, are 200 feet apart. And they're both looking at a tree at a nest in a tree. Uh, but we don't know the height of that tree, so we just call that why right now. But we do know is that the Closer Observer is looking up with an angle of elevation of 60 degrees. And the further away observer is looking up with angle of elevation of 30 degrees. And our goal was to find the distance from the observers to the base of the tree s. So what we want to do is use, um, write English trigonometry. We see that we have two right triangles here. The first is made with the closer observer, the nest and the base of the tree. The other is made with the further way observer a nest on the base of the tree. Right, So it seems like we don't know all that much information. We only know that these two observers or 200 feet away and we know they're angles, But we can use this information to, um, determine the distance of each observer from the tree. And tell Abe about distance. That's going to be for the closer observer X one this distance and for the further away observer, we can call it X two. But that's just gonna equal X one plus 200 right, because of the further away observer is 200 feet away from the closer to eso, we can use all this information. Teoh, find two equations and with two equations, which will have variables. Why an X one? Uh, we can solve for X one, and in doing so, we can also solve for X two. So let's set the situations up by the first equation. I'm going to redraw my green triangle here. The first equation is gonna have to do with his green triangle. Remember, our height is why here are adjacent side its X one, and this angle of elevation is 60 degrees. Make this, um and this is all the information that we know about this triangle. So it looks like we're gonna use a trigger metric function that relates the opposite and adjacent side. And that triggered metric functions. Tangents So we know that tangent of 60 degrees is equal to y over x one. Um, and that's as specific as we can get, yet We don't know anything about why. So we can't sell for X one. Um, we could rearrange this equation a little bit to make it a little bit more useful to us in the in the future. We know that, uh, well, I guess we'll get to that. We'll get to that a little bit later. Right now, examine the second triangle and make another equation here. Um, our second triangle is similar. It's not somewhere in a few Mr Sense, but we know some of the same information about it. We know that the height is why and that this angle is 30 degrees. This adjacent side is X two, but we said that that was equal to x one plus 200. So when we set up uh, the same type of equation we're gonna use Tangent, Charlie X one and R. J per Troy. Why? And are adjacent side are opposite and adjacent sides. We have a tangent this time of 30 degrees equals. Why? Over X one plus 200 Now we have two equations and we want Teoh, um, solve for X one. So the way that we might do this, we notice that each one of these has a why. So if we saw each equation for why, then we can set there's equal to each other. And, um, and use that information to sell for X one. So I'm gonna show what I mean. If tension of 60 degrees equals, why over X one? That means I can multiply both sides by x one, because I know X one is not equal zero. Um, and I get that X one times tangent of 60 degrees equals. Why? Um and I could do the same thing in this equation. The second equation, right. I can multiply both sides by X one plus 200 and I get the X one plus 200 times tangent of 30 degrees also equals Why eso I know why is the same in both triangles So I can set these equal to one another, right? So let's do that in this another line here. So what I have is that x one times the tangent of 60 degrees equals x one plus 200 times the tangent of 30 degrees. Now, luckily for me, I know with the tangent of 16. 30 degrees. All right, these are some of the special triangles that we know. Um, Ancient of 60 degrees is gonna be the square root of three. So it could rewrite this as one times X one times the square root of three and attention of 30 degrees is going to be its reciprocal one. Over, uh, scored of three eso. When I write that out, it looks like X one, plus 200 times one over the square to three. Now, we could we could have gone through this problem keeping the tangent of 60 intention of 30 degrees. Um, and just if we weren't familiar with these special triangles, just play this new a catchy eight at the end. Um, but I think it simplifies the, uh At least it makes the math look a little bit cleaner if we, uh, do that substitution right now. Um, so we have this and now we're trying to solve for X one. The way they were gonna do that is first we distribute this, we get this expression x one times escort of three equals, uh, X one over the square of three, plus 200 over the square to three. And I want to get all of my ex ones on both sides on one side. So I will subtract X one over the square to three from both sides, and I'll get X one times the score to three, minus x one over this word of three that's gonna equal 200. Well, 200. Excuse me. It's going to call 200 over the squirt of three. Um, so I'm what I want to factor a x one out of this term. So I get X one equals x one times the quantity scored a three minus one over the square to three. This looks confusing. So it's, um Let's kind of re express it. I know that I can re express one over the square root of three as, uh, squirt of 3/3. Or maybe I could instead find a common denominator. Um, yeah, I'm gonna do that instead. Actually, eso have one over the square of three. Now I want to find a common denominator. Seiken, subtract thes. The common denominator is gonna be the squared of three. Um, so this is going to be equal to X one Times square to three times the square to three over the square to three, minus one over the square to three. See, I've just multiplied. Uh, this squared of three, my first term in this binomial as by scored of three over the square of three, right? I just multiplied it by a special form of one to get this. And now when I simplify it, we see that my numerator scored three times in squared three. Well, that's just three, right? So three scored a three minus one of the square to three is equal to over the squirt of three. So I can re express this as, um, x one times two over the squirt of three. And remember, that is equal to 200 over the square to three. Right? That's we're still dealing with this with this equation. Cool. So now we have this, and now it's quite simple to solve for X one, we're gonna multiply both sides by the squirt of three, and what we get is x one times two equals 200. Write these words of three gonna cancel out. And now I divide both sides by two, and I get the X one equals 100. Perfect. So now that I know that X one equals 100 I think back to my diagram of here. Right, So I know that X one is 100. That means the closer observer is 100 feet away. And I know that x two, which is the distance away, that the further observer is from the tree, this just x one plus 200. So I know that the closer observer is 100 feet away and that the further observer or other observer is 300 feet away. No, this is kind of a long problem. So to recap, what we did was set up two right triangles, find two right triangles in our diagram, uh, so you could find two equations that way, even though we don't know the height, um, we can use the information from two equations in a something like a system of equations, right system of two equations to solve for X one. And in fact, once we know it's when we could go back and sulfur the height of the tree But that's not with the the problems asking. So we, uh, are satisfied just figuring out the X one equals 100 feet. And from that we can tell that X two equals 300 feet.

Ah, a surveyor is looking at a river, and he has a point. A. So we have. If River here and he is here at point A and there's a point across the river, which is now he is wondering what is the distance between Point A and Point B in order to determine that without crossing the river she is going to use. He's going to use another point here on the same side that he labels point see. And so they're going to measure from Point A to Point B, and that is the call for 240 feet. Now you see, we can catch off perhaps our triangle. So we're wondering again, What is this like from a to B Now? Because the opposite angle is labeled seat. We're going label that unknown side as lower case. She's Casey that were wanting Got the distance from Hated me eso. We've talked about this 1st 1 and labeled that so the angle across from A to B so a TV show is here, and the angle cross from that, which is angle C is equal to 62 degrees. Angle across from sea to bees, feed a thieve angle across from that single A that is equal to 55 degrees and then find the distance from A to B. We label that as R. C. So in order to use the law of science to solve for this, we need that third angle measure because half side like feels here. This if I'd like way need angle here. Triangle consists of 180 degrees. So in order to go to find that third angle should do 180 minus 60 to minus 5580 my five minus soon equals 63 so that is ankle be 63 degrees. So in order to find this length, we used lost signs, which says that we can have five blank be over spine of ankle be on. This will be equal to decide, like see over side angle to put numbers in silence. Busy to 40 Angle B is equal to 63 degrees equals two side like see, we don't know what it is. It does a very bold over sign of angle sea just 62 degrees. Now we need to solve for C, so we will more supply both sides, but Science 62 cancels these out. So we have 140 time sign of 62. Over 63 is equal to startling. See, so far costlier and pictures in degree mode. So we have 240. Multiply by sign, uh, 62 divided by a fine of 53. That equals 237. Quite So. Watch around this to a whole number. Um uh You look at this age, it's about five. We round up, we'll have 238 feet sidelight.

This is Problem number 36 in which we need to find the height of the kite of Okay, hard. Okay. It as describing the figure. So let us suppose height of the kite is at from the ground. This is ground. Okay, so they say hi to look at the 30. And one more thing we need to suppose that these distances x so a B and C that it's supposed a B and C so in triangle A no right angle triangle K B c tangent off 78 degree will be ach by acts that is perpendicular by base. So x will be at by 10 78 degree. Let us support this to be question number one. This is a question number one. We are getting a question number one in foreign term off it because we need to find the value Fetch not the value of X again in triangle right angled triangle K C 10 gent off 62 degrees will be called toe at buyer total a C, which is 30 plus x so, uh, actually be equal toe 10, 62 degrees in 2 30 plus x so 30 plus X will build on edge H by 10. 62 degree. This is a question number two. Question number two. Now, from a question number one and two, we have to plug in the value of fact from a question number one. So this will be 30 plus ach by 10, 78 degrees will be called toe ach by 10. 62 degree. Okay, so this can be done. Edge 30 equal to X Common 10, 62 degrees, minus one by 10. 78. Degree. Okay, so that's and we have to use the calculator calculator one over one development and 62 degree 10, 62 degrees minus one. Divided by 10. 78 degree, 0.3191 0.31 91 So height of the kite, which will be equal toe 30 by 0.31 91 30. Divided by 5.3191 94.1 94.1 for four. So height of the kite approximately will be called to 94 point 94.1 4 ft. They should be answered. Thank you so much.