A village p is 10km from a point x on a bearing 025degre from x another village Q is 6km from x on a bearing of 162degre.calculate the distance and bearing of p from Q.
Actually i want the answer to that question and how to solve it thanks.
To calculate the distance and bearing of village P from village Q, we need to use trigonometry and vector addition.
Step 1: Draw a diagram.
Draw a diagram with point X in the center. Draw a line segment towards the northeast (025 degrees) with a length of 10 km and label the endpoint as P. Now draw another line segment towards the southeast (162 degrees) with a length of 6 km and label the endpoint as Q.
Step 2: Calculate the coordinates of P and Q relative to X.
Using trigonometry, we can calculate the coordinates of P and Q relative to X.
For P:
x-coordinate of P = 10 km * cos(025 degrees)
y-coordinate of P = 10 km * sin(025 degrees)
For Q:
x-coordinate of Q = 6 km * cos(162 degrees)
y-coordinate of Q = 6 km * sin(162 degrees)
Step 3: Calculate the distance between P and Q.
Using the coordinates of P and Q, we can apply the distance formula:
Distance (PQ) = √[(x-coordinate of P - x-coordinate of Q)^2 + (y-coordinate of P - y-coordinate of Q)^2]
Step 4: Calculate the bearing of P from Q.
Using the coordinates of P and Q, we can determine the bearing of P from Q.
Bearing (P from Q) = arctan[(y-coordinate of P - y-coordinate of Q) / (x-coordinate of P - x-coordinate of Q)]
Step 5: Calculate the final result.
Plug in the values into the formulas and calculate the distance and bearing of P from Q.
To calculate the distance and bearing of point P from point Q, we can use the principles of trigonometry and vector addition.
First, let's visualize the problem. Draw a diagram with points P, Q, and X as mentioned in the question. The bearing of 025 degrees from X means that the angle between the lines PX and X-axis is 25 degrees. Similarly, the bearing of 162 degrees from X means that the angle between the lines QX and X-axis is 162 degrees.
To calculate the distance and bearing of P from Q, we need to find the length of line QP and the angle it makes with the X-axis. Here are the steps:
1. Calculate the coordinates of points P and Q relative to X:
- Let the coordinates of X be (0, 0) for simplicity.
- Given that P is 10 km away at a bearing of 025 degrees, we can use trigonometry to find its coordinates.
- The horizontal distance from X to P is given by 10 km * cos(25 degrees).
- The vertical distance from X to P is given by 10 km * sin(25 degrees).
- Similarly, for Q at a bearing of 162 degrees, the coordinates can be found using trigonometry.
- The horizontal distance from X to Q is given by 6 km * cos(162 degrees).
- The vertical distance from X to Q is given by 6 km * sin(162 degrees).
2. Calculate the distance QP (d) between points Q and P:
- Use the distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2), where (x1, y1) and (x2, y2) are the coordinates of Q and P, respectively.
3. Calculate the bearing (θ) of point P from Q:
- The bearing can be calculated using the arctan2 function or by taking the inverse tangent of the change in y divided by the change in x.
- The bearing should be in the range of 0 to 360 degrees. If the resulting angle is negative, add 360 degrees.
After performing these calculations, you will have both the distance and bearing of point P from Q.
you can find the distance PQ using the law of cosines:
x^2 = 10^2 + 6^2 - 2(10)(6)cos137°
x = 14.959
for the bearing, locate P and Q in rectangular coordinates and then the bearing is
90-arctan(y/x)
where x and y are the corresponding displacements from Q to P.