A man starts at p and travels 5km on a bearing 040 degree to point q he then travels 5km on a bearing 210 degree to r find the bearing and distance of r from p
Following your description you should get a triangle with sides 5 and 5 and an angle of 20° between them, so use the cosine law:
r^2 = 5^2 + 5^2 - 2(5)(5)cos20
= 3.01536...
r = 1.7364..
could have dropped a perpendicular and then
sin10 = (r/2) / 5
r = 1.7364... (notice the same answer, that's encouraging)
you now have enough data to find the missing angles
Hard question ❓
To find the bearing and distance of point R from point P, we can break down the journey step-by-step.
Step 1: Starting at point P, the man travels 5km on a bearing of 040 degrees to reach point Q.
Step 2: Next, the man travels 5km on a bearing of 210 degrees from point Q to reach point R.
Now let's calculate the final bearing and distance from P to R.
Step 1: From point P to point Q:
- Distance: 5km
- Bearing: 040 degrees
Step 2: From point Q to point R:
- Distance: 5km
- Bearing: 210 degrees
To calculate the final bearing and distance, we need to add the bearings:
040 degrees + 210 degrees = 250 degrees
To calculate the final distance, we need to use the cosine rule:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where:
- c is the final distance (PR)
- a is the distance from P to Q (PQ) = 5km
- b is the distance from Q to R (QR) = 5km
- C is the angle between a and b (the difference between the bearings) = 170 degrees
Plugging in the values:
c^2 = 5^2 + 5^2 - 2(5)(5) * cos(170 degrees)
c^2 = 25 + 25 - 50 * cos(170 degrees)
c^2 = 50 - 50 * cos(170 degrees)
Using a calculator:
c^2 = 50 - (-0.984808)
c^2 = 50 + 0.984808
c^2 = 50.984808
c ≈ √50.984808
c ≈ 7.14 km
Therefore, the bearing from P to R is 250 degrees, and the distance from P to R is approximately 7.14 km.
To find the bearing and distance of point R from point P, we can use vector addition and trigonometry. Here's how:
1. Start by drawing a diagram to visualize the problem. Mark point P as the starting point, and label the bearings to point Q (040 degrees) and point R (210 degrees).
2. Calculate the displacement vectors from point P to point Q and from point P to point R. Since both distances are 5km, the displacement vectors for both cases will be 5km in length.
3. To find the components of the displacement vector from point P to point Q, we can use trigonometry. Since the bearing is given as 040 degrees, we can break down the vector into its x and y components.
The x-component (Qx) can be calculated using the cosine of the angle: Qx = 5km * cos(40 degrees).
The y-component (Qy) can be calculated using the sine of the angle: Qy = 5km * sin(40 degrees).
Note: Make sure to convert the angle to radians if your trigonometric functions require radians instead of degrees.
4. Repeat the same process for the displacement vector from point P to point R. Since the bearing is given as 210 degrees, the components are:
Rx = 5km * cos(210 degrees)
Ry = 5km * sin(210 degrees)
5. Now, add the x and y components of the displacement vectors together to find the overall displacement from point P to point R. This can be done by summing the x-components and the y-components separately:
Total displacement in the x-direction (Dx) = Qx + Rx
Total displacement in the y-direction (Dy) = Qy + Ry
6. Using the total displacement, find the distance (R) between point P and point R using the Pythagorean theorem:
R = sqrt(Dx^2 + Dy^2)
7. Finally, calculate the bearing (B) of point R from point P. The bearing is the angle between the displacement vector and the north direction. This can be found using trigonometry:
B = arctan(Dy / Dx) in degrees
Note: Again, make sure to convert the angle to degrees if required by your trigonometric functions.
8. Now you have the bearing and distance of point R from point P.