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Heights and distances word problem: distance between two buildings

Let's solve a problem in real-time involving finding the height of a tall building given the angles of depression of the top and bottom of another shorter building. Created by Aanand Srinivas.

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Video transcript

it's story time the angles of depression of the top and the bottom of an 8 meter tall building 8 meter tall building so let me just let me just draw that I have an 8 meter tall building over here and what I know is that what do I know I know that this building is eight meters tall that that's all I know for now from the top okay the angles of depression of the top and the bottom the top and the bottom of an eight meter tall building from the top of a multi-storied building okay now it makes sense so there is a multi-story building somewhere and it's it's clearly taller because so why is it taller actually yeah why is it clearly taller it's clearly taller because there are angles of depression for both the top and the bottom of this 8 meter tall buildings so this is the top of the 8 meter tall building this is the bottom and both have angles of depression if this building had been higher then from from here one of the points would have had an angle of elevation and the other point would have had an angle of depression so okay both have an angle of depression so this is a taller building and maybe are also helping us a multi-storied you know to get the hint it's a tall building or 30 and 45 degrees okay so now I need to draw this part this is the interesting part what is where do I draw this 30 and 45 in fact once I do this the question probably hopefully becomes easier so what is this the angle of depression of the top from the top of her okay so not from somewhere here but somewhere here from the top of a multi-storied building at 30 and 45 so the horizontal is going to be this that's the horizontal now because the angle of depression is at 30 for the top and as he respectively over here which means that it's 30 for the top and 45 for the bottom it would be really weird for it to be the other way so in anyway this is they may not even have given that so oh no not this red line over here yeah for the top it's 30 degrees which means that this angle is 30 degrees and let's see now I need took it to go to the bottom so this angle this line is the line of sight to the bottom and that angle will be 64 60 degrees so no what is it it's 45 degrees 45 degrees got it that's all that's given to me now I need to know okay what do you want what is the question want find the height of the multi-story building oh this height is what we want this height let me mark that whenever I don't know the name of something that's going to give it a name that this height is what I want and the distance between the two buildings the distance that is this one over here this distance is the other thing that we want now this is usually the part of the question that we need to be careful about because now once we have this it boils down to asking where are the right triangles here now as I look one thing I can see is because this is 30 all right I know that I need it I need to find the right triangle and I know that because this angle is given to me it's a clue for me to draw maybe this or basically imagine in my head this line because this line is not a real line right this is this is an exist but I imagine that and I know that this will also be 30 which gives me a triangle over here can you see that a right triangle over here and similarly there's another much more real right triangle over here because this is the height of the building and the multi-story building and this is the distance D so this is much more real and that angle will be 45 degrees wait a minute if that is the case if the bottom is 45 degrees then find the height of the multi-story building and the distance between the two so are to find two things but this must mean that both these are the same because 45 degrees means that its opposite side and the adjacent side will basically be equal right because this is 45 this is 90 this will also be 45 which means that dnh are actually the same thing so we're finding two things but it's actually just one thing makes me feel better now how do I do this though the first thing I can see is the only length given to me is 8 so before I even start questions like this what I like to do is ask hey is this even findable like what I call you notices is this problem solvable and the way I like to think about it is if I I'm standing on the building over here and if I have one angle of depression line like basically if I shoot a laser at a 30-degree angle below the horizontal and if I shoot another laser at 45 below the horizontal then these two lasers can meet it can can keep going forever but there's only one point at which they'll make a distance they'll have a distance eight between them all points before this they'll have a lesser distance between them and all points after there's a level larger distance because they're always diverging so I know that with the information given to me I can actually pinpoint this point over here start at 8 go back you will find this point because finding this point is exactly the same as finding this distance and this height you can do that so now what do I do okay because I know this height is 8 i-i'll pick this height is also 8 so this length over here but that's not directly the side of any triangle what I'm really looking for is a way to relate things that I know to the things that I want and I know because this is D this is D this will also be D this will also be D let's color that so that you know what I'm talking about that'll also be D and this side we don't know what that is right but this will be basically if I just have to like write it over here I'll maybe say it's H minus 8 it's minus 8 because this whole thing is H this is 8 so this is H minus 8 I think I have I think I I I yeah I think I know what to do here let's see I have two variables basically H and D which I need to find which because it's 45 degrees I kind of can even eliminate already I can just call both of them the same thing but certainly I think I should I should do for now but if that wasn't the case if this was some some other angle however thought about it is there are two variables two equations the two equations will come from one from this triangle by relating this side to this side using the tan of this angle and the other equation comes from this triangle relating this side and this side using the tan of this angle so let's just do the 45-degree one first because you can see clearly that this is H this is D so H by D will be equal to tan 45 opposite by opposite by adjacent and an 45 is equal to one but I don't know if you should remember that so you can always just I always just draw it very quickly for myself and I have 45 over here and I know if this is one then this will be one because it's an isosceles triangle and this will be root 2 by Pythagoras theorem so here I need tan so it's 1 basically that gives for me h equals d that is super convenient makes the question much simpler so i can forget all the DS that I see or or maybe I'll forget the H is that I see and then market mark them as DS and I know that that's all I'm trying to find now the second thing I need to do is look at this triangle so i know h minus 8y D will be equal to tan 30 right opposite by adjacent equals tan of this angle so H minus 8 this is where the 8 is coming in handy this information it's needed by D equals R tan of 30 degrees turn of 30 degrees and to remember tired of 30 once again I draw my what I call or I like to call my 60 degree triangle so let me draw that say o say maybe over here so I have my 60 degree triangle that does not look like 60 degrees so 60 degree triangle and I know that if I draw this then if this is 1 then this will be 2 and this length over here will be root of 2 square minus 1 square or root 3 so tan 30 is root 3 by 1 tan 30 oh no tan 30 is over here this is 30 so tan 30 will be 1 by root 3 no it's easy to get and get a little bit confused but I always like to think he's a smaller angle right so it's tan can't be root 3 so it's done has to be less than 1 so it's 1 by root 3 so with that I would have hold on a little bit over here so 8 minus 8 by D by D will be equal to tan 30 or 1 by root 3 and this gives me so I have one equation one variable oh I forgot to replace my H with D I for to have one equation I should have used this idea so it's D minus 8 by D equals 1 by root 3 and now it becomes do I do this hmm so maybe I will I'm just quickly thinking whether I should keep the route three on this side or move it there we do all of this and some of some of these pre calculations and I head right to minimize work lazy people so I'm gonna just maybe cross multiply so left D route 3 maybe I can keep the color so that you know where things are moving so this route 3 I'm just multiplying on this side so D root 3 minus 8 root 3 8 into root 3 will be equal will be equal to D right now my job is to just take collect the DS to one side which is basically the same as subtracting by D on both sides of the equation so I love D D times root 3 minus 1 root of 3 minus 1 and that'll be equal to 8 root 3 8 into root 3 C now you should notice that actually the part where you're using anything to do with trigonometry is over long ago like that was that was over when you wrote these two equations tan 45 and tan 30 the part where you are dealing with heights and distances was over even before that when you drove through this diagram which is probably the the interesting part converting the story into the into the right triangles now we are basically doing the algebra needed to give an answer that doesn't look too you know too ugly you want to make the answer as beautiful as possible so you have D equals 8 root 3 by root 3 minus 1 and almost always I leave it there but many times the teacher used to insist that I do this thing called rationalizing the denominator which is basically just we don't like for some reason I have no idea why we don't like square roots in the denominator so what you do to get rid of that is you treat this is a minus B and then you put a plus B on both sides on the numerator and denominator and if you do that the reason you do this is you'll get a square minus B Square you'll get rid of the square roots so this will be equal to 8 root 3 into root 3 plus 1 which should be something let's go that but the bottom is simpler so root 3 minus root 3 into root 3 is 3 minus 1 will be 2 so you have a nice neat little 2 below and you have an 8 over here eight over here and you have root 3 into root 3 plus one I'm just gonna carry this inside in and multiply you'll have 3 plus root 3 right that's what you'll have 8 into 3 plus root 3 so let's just cancel the 8 with the 4 over here into the 2 and get a 4 over here so let's say 4 & 4 into 3 plus root 3 seems to be the answer for D hmm I don't know whenever I get answers like this I feel a little bit weird maybe I should calculate the final answer maybe I should leave it here so 4 into 3 plus root 3 will be equal to 12 plus 4 root 3 hmm okay maybe that's the answer maybe I should be accepting answers that look a little bit non-standard let's say so then I'll come here and I'll say let's move this a little bit back over here so D equals 4 into 3 plus root 3 and then this D will also be the same thing the height is also equal to 4 into 3 plus root 3 and we have solved the question