A village r is 10km from a point p on a bearing 025degree from p. Another village A is 6km from p on a bearing 162degree.calculate distance of r from a and the bearing of r from a.
From my diagram, I have triangle PRA with angle P at 137°, PR = 10 and PA = 6
AR^2 = 6^2 + 10^2 - 2(6)(10)cos137°
I get AR = 14.96 km
sinR/6 = sin137/14.96
R = 15.875°
bearing = 25-15.875 = appr 9.1°
or by vectors:
vector AR = (10cos25,10sin25) - (6cos162,6sin162)
= (14.769... , 2.372..)
magnitude = √(14.769..^2 + 2.372..^2) = 14.956 , same as using the cosine law.
tanØ = 2.372/14.769
Ø = 9.1°
All angles are measured CW from +y-axis.
AR = AP+PR = 6[162+180] + 10[25o].
AR = (6*sin342+10*sin25) + (6*cos342+10*cos25)i
AR = 2.37 + 14.8i = 15km[9.1o].
To find the distance between village R and village A, as well as the bearing of village R from village A, we can use trigonometry and basic geometry.
1. Start by drawing a diagram. Draw a point P and label it. Then draw a line segment 10 km long extending in the direction of bearing 025 degrees. Label the endpoint of this line segment as village R. Finally, draw another line segment 6 km long extending in the direction of bearing 162 degrees from point P. Label the endpoint of this line segment as village A.
2. To find the distance between village R and village A, we can use the Law of Cosines. The Law of Cosines states that in any triangle, the square of one side is equal to the sum of the squares of the other two sides, minus twice the product of the magnitudes of those two sides multiplied by the cosine of the included angle. In this case, we can consider triangle PRA, where PR is the side opposite the angle at village A, PA is the side opposite the angle at village R, and RA is the side opposite the angle at point P.
Using the Law of Cosines, we have:
RA^2 = PR^2 + PA^2 - 2 * PR * PA * cos(angle RPA)
We know that PR = 10 km and PA = 6 km. To find the angle RPA, we can subtract the bearing of R from the bearing of A: angle RPA = 162 degrees - 25 degrees = 137 degrees.
So the equation becomes:
RA^2 = 10^2 + 6^2 - 2 * 10 * 6 * cos(137 degrees)
Now you can calculate RA by plugging in the values and using a calculator.
3. To find the bearing of R from A, we can use the concept of angles in a triangle. We know that the sum of the angles in a triangle is 180 degrees. In triangle PAR, we have:
angle APB + angle PBA + angle BAP = 180 degrees.
Since angle APB is 90 degrees (as PA is perpendicular to PB), and angle PBA is 137 degrees (as calculated earlier), we can solve for angle BAP, which will give us the bearing of R from A.
angle BAP = 180 degrees - 90 degrees - 137 degrees
Now you can calculate the value of angle BAP, which will give you the bearing of R from A.
Once you have the distance and the bearing, you can state the final answer as follows:
The distance between village R and village A is [RA] km, and the bearing of village R from village A is [angle BAP] degrees.