Two geological field teams are working in a remote area. A global positioning system (GPS) tracker at their base camp shows the location of the first team as 38 km away, 19deg north of west, and the second team as 28.7 km away, 35deg east of north. When the first team uses its GPS to check the position of the second team, what does the GPS give for the second team's distance from them?
my answer is 53.6km, i now need to find the direction in degrees measured from due east
I just saw related questions so I figured it out.
(28.7/53.56)(sin 106) = 0.515
sin-1 0.515= 31.00deg
31.00deg -19deg = 12.00 deg
Law of cosines:
c^2=28.7^2+38^2 - 2(28.7*38)cosAngle
and the angle is 71+35
so that give you the magnitude.
Now, the direction. Law of sines
SinTheta/28.7=sinAngle/magnitude
and the direction is Theta-71
draw all this out on a sketch and verify.
To find the distance between the two teams, you can use the law of cosines, which states that in a triangle, the square of one side is equal to the sum of the squares of the other sides minus twice the product of their lengths and the cosine of the angle between them.
Let's label the distance between the first team and the base camp as A, the distance between the second team and the base camp as B, and the angle between them as θ.
Using the given information:
A = 38 km (distance of the first team from the base camp)
B = 28.7 km (distance of the second team from the base camp)
First, we can find the third side of the triangle formed between the two teams and the base camp. Let's call it C.
C² = A² + B² - 2AB * cos(θ)
θ = 35° + 90° (since the angle measured east of north needs to be converted to an angle measured from due north)
θ = 125°
C² = 38² + 28.7² - 2 * 38 * 28.7 * cos(125°)
C² ≈ 1444.24
C ≈ √1444.24
C ≈ 38.01 km
Therefore, the GPS will show that the second team is approximately 38.01 km away from the first team.
To find the direction or bearing in degrees measured from due east, we can use trigonometry. Let's call the angle we want to find α.
sin(α) = A / C
sin(α) = 38 / 38.01
α ≈ arcsin(38 / 38.01)
α ≈ 89.999°
Since the first team is west of the second team, the bearing is measured from due east. The angle we calculated above is 89.999°, which is just a tiny bit less than 90°.
Therefore, the GPS will show that the second team is approximately 38.01 km away from the first team, with a bearing of approximately 90° (measured from due east).