 In this video, we provide the solution to question number 13 for the practice exam number one for math 1060, in which case I'm just gonna read it real quick. To estimate the height of a tree, one person positions himself due south of the tree while another person stands due east of the tree. If the two people are the same distance from the tree and 35 feet from each other, what is the height of the tree if the angle of elevation from the ground at each person's position to the tree, top of the tree is 48 degrees? So there's a lot going on in this story problem. It can be very helpful to try to draw a picture to help us illustrate what's going on here. So think of about the following situation here. So we have these two axes and we're gonna draw a tree which I don't have the color brown available to me. So our tree is just gonna be yellow. That's the closest color I can get to a tree. We'll put some little cute little leaves on top. There's our tree, great. And so we have a person who walked south of the tree and someone who walked east of the tree. So if we say that this direction is south, therefore this would be north. And this one would, we'll call this one east and this one is west. We feel weird that north isn't at the top. Maybe I'm a dwarf, maybe I just, I just didn't think about it beforehand. It doesn't really matter. This is how our situation is set up. And so our people walked some distance from the tree. We don't know what that distance is, but we do know they went the same distance. So the person who went east and the person who went south, they went the same distance away from the tree. It turns out we don't actually need to know what that is. But what we do know is that the distance between the two people is 35 feet, that we know. And also we know that the angle, the angle of elevation to the top of the tree, that's gonna turn out to be 48 degrees, 48 degrees. I'm gonna draw that triangle right here as well, 48 degrees. And so again, we don't know the distance to the tree, let's call that D. Maybe it is something we wanna figure out at some point, but just, we're kinda drawing this picture right here to kinda get an idea of what do we know, what do we wanna know. So we definitely need to know the height of the tree. So if we were to draw some of these triangles separately, so one of the triangles we have is the follow and there's some distance D, it's a right triangle. There's some distance H, and we have this angle of elevation of 48 degrees. So if we just focus on that, so that's just this triangle right here. If we just focus on that triangle for a moment, just using the tangent ratio, we get H over D is equal to tangent of 48 degrees. Like so, solving for H, we're gonna get H is equal to D times tangent of 48 degrees. Okay, so if we knew the distance, the two people walked away from the tree, we could find the height of the tree. That's often how you do these measurements. But since we don't know the height, excuse me, since we don't know the distance they've walked away, that's the reason why there's two people, we know the distance between them, the 35 feet. How can we utilize that fact whatsoever? Well, it turns out that there is another right triangle right here in play, for which what we know about this is there's 35 feet right here, and we also know there's a distance D and D, like so. So we could solve this using the Pythagorean equation. We could solve for D, we're gonna get D squared plus D squared is equal to 35 squared. In other words, you get two D squared equals 35 squared. D squared is gonna equal 35 squared over two, take the square root, you get D equals 35 over the square root of two. So you could use the Pythagorean equation to do this, but another observation I wanna point out to you is that this triangle here is necessarily the Saucely's right triangle, how do I know that? Well, because this is D, this is D, the two legs are congruent to each other, so that makes it a Saucely's triangle, this has to be a 45 degree angle. And so we could use, we could in fact use the trigonometry of a 45, 45, 90 degree triangle. So in other words, the legs are just gonna be the hypotenuse divided by the square root of two. So D equals 35 over the square root of two. So if you had recognized it was the Saucely's triangle, you could have skipped some of the work necessary to get 35 over the square root of two, but nonetheless, it's gonna be the same answer no matter how you approach it. So putting this information into this equation right here, we're gonna get H equals 35 over the square root of two times the tangent of 48 degrees. Now, when it comes to these problems, you always need to give me an exact answer. Honestly, the approximations I don't really care about. This is the calculator portion of the test, but you'll see that there actually was nothing that required a calculator here. If you wanna do an approximation, that is appropriate, but for full credit, give me the exact answer. The approximation put into your calculator would be 27.486, it doesn't say how many decimal places to put it because it actually doesn't ask for it. So we'll just say that the tree is approximately 27 and a half feet tall.