A girl moves from a town M on a bearing of 065° to a town N, 100km away. She then moves from N on a bearing of 130° to a position P, 200km from N. find the :(a). The distance between M and P. (b). The bearing of M from P correct to the nearest degree
Using the diagram which I assume you made, you should see that, after a bit
of basic geometry, for triangle MNP the angle at N = 115°
And by the cosine law:
MP^2 = 100^2 + 200^2 - 2(100)(200)cos115°
I got MP = appr 258.66 km
by the sine law we find angle M to be ...
sinM/200 = sin115/258.66
angle M = 44.5°
add on you 65° and you have a bearing of 109.5°
or , by vectors
bearing of 65° ---> standard angle of 25°
bearing of 130° ---> standard angle of -40°
resultant = 100(cos25, sin25) + 200(cos-40, sin -40)
= (90.63, 42.26) + (153.21, -128.56)
= (243.84, -86.296)
|resultant| = √(243.84^2 + (-86.296)^2 ) = 258.66 km, as above
direction of resultant:
tan 𝛳 = -86.296/243.84
𝛳 = -19.5°
so bearing = 90+19.5 = 109.5° , as above
draw a diagram
then use the law of cosines to find MP
use the law of sines to find the angles at M and P
then you calculate (b)
come back if you get stuck, and show your work
To find the distance between town M and position P, you can use the concept of vector addition. We will break down the problem into smaller steps and use trigonometry to find the answer.
(a) The distance between M and P:
1. Start by finding the coordinates of the towns M, N, and P on a coordinate plane. Let's consider town M as the origin (0,0) for simplicity.
2. Since the girl moves on a bearing of 065° from M to N, it means she moves in the northeast direction. To find the coordinates of N, we can use trigonometry.
- The horizontal displacement from M to N is given by cos(065°) * 100 km.
- The vertical displacement from M to N is given by sin(065°) * 100 km.
- Therefore, the coordinates of N will be (cos(065°) * 100, sin(065°) * 100).
3. Next, the girl moves from N on a bearing of 130° to position P. Similarly, we can calculate the coordinates of P using trigonometry with respect to N.
- The horizontal displacement from N to P is given by cos(130°) * 200 km.
- The vertical displacement from N to P is given by sin(130°) * 200 km.
- Therefore, the coordinates of P will be (cos(130°) * 200 + N's x-coordinate, sin(130°) * 200 + N's y-coordinate).
4. Find the distance between M (0,0) and P (computed coordinates). You can use the distance formula:
- Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2), where (x1,y1) = (0,0) and (x2,y2) are the coordinates of P.
- Calculate the values and you will get the distance between M and P.
(b) The bearing of M from P (correct to the nearest degree):
1. To find the bearing, we need to determine the angle between the line connecting M and P and the horizontal axis.
2. Calculate the horizontal displacement from M to P, given by P's x-coordinate - M's x-coordinate.
3. Calculate the vertical displacement from M to P, given by P's y-coordinate - M's y-coordinate.
4. Use the formula for tangent to find the angle in degrees: tan^(-1)((vertical displacement)/(horizontal displacement)).
- This will give you the angle in radians. Convert it to degrees.
- Round the final result to the nearest degree.
By following these steps, you can find both the distance between M and P and the bearing of M from P.