00:01
In the given question we are told that points a and b are 100 meters apart on a cost line.
00:08
So b is directly east of a and a lighthouse is out at the c.
00:15
So we have a figure along with this question which illustrates this problem and we have the lighthouse over here.
00:24
The point b is about here and a is over here.
00:30
These are the distances from the lighthouse to the points a and b.
00:38
The distance between a and b is given as 100 meters.
00:43
And now what we are told us b is directly east of a.
00:50
So the directions are north as upwards, south is down.
00:55
Towards east is towards the right and west is towards the left so in this figure we can see that b is to the east of a and now we are told the bearing of the lighthouse is 164 degrees from a and 215 degrees from b so what we could do over here is let's mark these angles and what else have been said to us is that the angle of elevation to the top of the lighthouse from the point a is 27 degrees and first what we have to find is the distance of the lighthouse from a.
01:49
That is this distance is what we have to find so what we can start with over here is we can see that we can draw this illustration as a triangle over here triangle like this yeah so we can draw a triangle like this and we can draw a triangle like this and we can just draw two vertical lines at a and b so that we can show the bearings from a and b towards the lighthouse so if this point is a and this point is b and this is the lighthouse we are told that the bearing from the point a is 164 degrees and the bearing from the point b that is we should measure it in the clockwise direction so the bearing at the point b is given us 217 right 200 and 15 degrees and now we are told first we have to find the distance of the lighthouse from a so what we can do in this case is we can take the angle we can first take the find the angles inside this triangle so if this is the angle a this is the angle b or let's take these as alpha and beta so we can simply write from this figure that since this is 90 degree these two are 90 degrees what we could find is that when we take alpha we can find it as 164 minus 90 degrees which would be equal to 74 degrees and for beta and for beta what we should know is that this entire angle that is this entire angle to the horizontal over here is to 70 degrees so which means in order to find beta we can subtract 270 degrees the bearing from the 270 degrees and we would have 270 minus 250 is 55 degrees so this is the value of beta now that we have the angles inside this triangle the third angle over here can be easily found by subtracting the other two angles from 180 because the sum of interior angles in a triangle in a triangle is equal to 180 degrees which means angle l is equal to 180 degree minus 74 degrees minus 55 degrees so when we take this when we do this that is 180 minus 74 minus 55 what we get as the angle measure of angle l is 51 degrees now that we have these angles we can write the distance between a and b is a hundred meters the angles are all the angles are found so now we have a simple triangle in which we have the lighthouse over here.
06:00
This angle is 51 degrees.
06:03
The angle over here that is at b is 55 degrees and angle at a is what? it is 74 degrees, right? and the distance between a and b is 100 meters.
06:18
Now we have to find the distance between a and l.
06:22
So what we can do is we can write.
06:26
The law of signs the low of signs by which it says that if we have a triangle of the form a b c with length of sides small a small b and small c and we have the when we use the law of signs in this law of signs in this triangle what we could write is that a divided by sign of the angle a is equal to b divided by sine of the angle b is equal to c divided by sign of the angle c...