00:01
In the given question we are told that wichita in kansas is due north of the place fort worth in texas.
00:11
And the following figure is given to us as well.
00:16
And we are told that this means that they lie on a circle whose cinder is that of the earth and whose radius is equal to the polar radius of earth.
00:25
And in the figure we can see that a distance is given to us over here and this is what we are told to take as the radius of earth 3 ,950 miles, right? and then we are told that the latitude of wichita is about 37 degrees north which is marked in the figure as you can see and the latitude of fort worth.
00:55
Is given to be 32 degrees north and now what we are asked to find is the distance between wichita and fort worth so what we could do over here first is we can take this arc over here this arc and this would be the distance between wichita and fort worth right so what we are going to do is we have considered earth as a circle and now we are told that bichita is 37 degrees north and fort worth is 32 degrees north.
01:33
So this arc length, let's say this angle would be this angle would be 37 degrees minus 32 degrees which would be 5 degrees.
01:49
So we can separately draw this arc, draw this sector.
01:55
And what we can mark is that this angle over here is what we have found as 5 degrees.
02:05
And this is between wichita, which is 37 degrees due north, and fort worth, which is 32 degrees due north.
02:20
Right so now what we have to find over here is this distance right this distance is what we have to find over here and we already know that the distance that we can take as the radius of this sector is the radius that we can take of the sector is 3 ,950 miles right so now we have changed this problem to a typical problem where we are just finding the arc length and to find the arc length we can use a simple formula which is arc length is equal to arc length is equal to the radius times the central angle of the sector so over here the radius is 3 ,950 miles and when we take the central angle, we take it in terms of radiance.
03:29
So in this formula, we take the angle in terms of radiance, but what we have found the measure of the angle to be is in terms of degrees, right? so we should change the degree measure to radiance first, and to do that, we can use this conversion where we already know that 180 degrees can be taken as pi radiance, right? so if this is so, we can take one degree as pi by 180 radian, right? and then we can write, we can just multiply 5 on both sides of this equation.
04:08
And we can write 5 degrees as then is then equal to 5 times pi by 180 radiant...