Two normads leaves camp at the same time,one travel 5km/h on a bearing of 39 degree and the other travel 7.5km/h on a bearing of 265 degree.how far are apart and what is the bearing of the second from the first?
"how far are apart" ?
After what time ?
Let the time passed since they left be t hours
My diagram shows a triangle with sides 5t and 7.5t and an angle of 134° between those sides.
If the distance between them is x, by the cosine law:
x^2 = (5t)^2 + (7.5t)^2 - 2(5t)(7.5t)cos134°
= 25t^2 + 56.25t^2 - (-52.01 t^2)
x^2 = 133.35 t^2
x = √133.35 t
x = appr 11.55 t km
To find the distance between the two nomads and the bearing of the second nomad from the first, we can use the concept of vectors.
Let's refer to the first nomad's velocity as V1 and the second nomad's velocity as V2.
The velocity V1 can be broken down into its horizontal (V1x) and vertical (V1y) components using trigonometry. The horizontal component is V1x = V1 * cos(39°) and the vertical component is V1y = V1 * sin(39°).
Similarly, for the second nomad, the horizontal component is V2x = V2 * cos(265°) and the vertical component is V2y = V2 * sin(265°).
Now, let's find the displacement, which is the difference between the positions of the two nomads.
The horizontal displacement (Dx) is the difference between the horizontal components: Dx = V2x - V1x.
The vertical displacement (Dy) is the difference between the vertical components: Dy = V2y - V1y.
The distance between the two nomads (D) can be found using the Pythagorean theorem: D = sqrt(Dx^2 + Dy^2).
Finally, we can calculate the bearing of the second nomad from the first (θ) using the atan2 function: θ = atan2(Dy, Dx).
Let's substitute the given values and calculate the result:
V1 = 5 km/h, V2 = 7.5 km/h
V1x = 5 * cos(39°) ≈ 3.819 km/h
V1y = 5 * sin(39°) ≈ 3.200 km/h
V2x = 7.5 * cos(265°) ≈ -5.722 km/h
V2y = 7.5 * sin(265°) ≈ -3.250 km/h
Dx = -5.722 km/h - 3.819 km/h ≈ -9.541 km/h
Dy = -3.250 km/h - 3.200 km/h ≈ -6.450 km/h
D = sqrt((-9.541 km/h)^2 + (-6.450 km/h)^2) ≈ 11.438 km
θ = atan2(-6.450 km/h, -9.541 km/h) ≈ -141.078°
Therefore, the two nomads are approximately 11.438 km apart, and the bearing of the second nomad from the first is approximately 141.078° (measured clockwise from the east).