The bearing of P from Q is 140degree and that of X from P is 315degree. If |PQ|=36km, and |PX|=55km, find |QX| and the bearing of Q from X.
since Q to P is 140°, P to Q is 320°
P to X is 315°, so angle QPX is 5°
using the law of cosines,
|QX|^2 = 36^2 + 55^2 - 2*36*55*cos5° = 376.1
So, |QX| = 19.39 km
55^2 = 19.39^2 + 36^2 - 2*19.39*36*cosQ
So, Q = 165.7°
The bearing of X from Q is thus
140+165.7 = 305.7°
sorry - you wanted the bearing of Q from X. Just subtract 180°.
To find |QX|, we can use the Cosine rule. The Cosine rule states that for a triangle with sides a, b, and c, and angle C opposite to side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we know that:
- |PQ| = 36 km, which is the side opposite to angle P,
- |PX| = 55 km, which is the side opposite to angle X, and
- The angle between |PQ| and |PX| is 315 degrees - 140 degrees = 175 degrees, which we'll call angle QPX.
Applying the Cosine rule, we have:
|QX|^2 = |PQ|^2 + |PX|^2 - 2 * |PQ| * |PX| * cos(QPX)
Since we know all the values, we can substitute them in:
|QX|^2 = 36^2 + 55^2 - 2 * 36 * 55 * cos(175)
Now, we can calculate |QX|:
|QX|^2 = 1296 + 3025 - 3960 * (-0.087)
|QX|^2 = 396 + 3960 * 0.087
|QX|^2 = 396 + 344.52
|QX|^2 = 740.52
Taking the square root of both sides, we find:
|QX| ≈ 27.2 km
To find the bearing of Q from X, we need to calculate the angle QXP using the Sine rule. The Sine rule states that for a triangle with sides a, b, and c, and angles A, B, and C opposite to sides a, b, and c, respectively, the following equation holds:
sin(A) / a = sin(B) / b = sin(C) / c
In this case, we know that:
- |PQ| = 36 km, which is the side opposite to angle Q,
- |PX| = 55 km, which is the side opposite to angle P, and
- |QX| ≈ 27.2 km, which is the side opposite to angle QXP.
Using the Sine rule, we have:
sin(Q) / |PQ| = sin(X) / |PX| = sin(QXP) / |QX|
We are looking for the angle QXP, so we can rearrange the equation as follows:
sin(QXP) = (sin(Q) * |QX|) / |PQ|
Now we can substitute the known values and calculate the sine of angle QXP:
sin(QXP) = (sin(140) * 27.2) / 36
sin(QXP) = (0.985 * 27.2) / 36
sin(QXP) ≈ 0.734
To find the bearing of Q from X, we need to convert the sine value back into an angle. We can use the inverse sine function (also known as arcsine) to do this:
QXP ≈ arcsin(0.734)
QXP ≈ 47.8 degrees
Therefore, the bearing of Q from X is approximately 47.8 degrees.