Final answer:
The distance from village Y to village Z, given Y's bearing from X and that Z is directly east of X, is approximately 17.3 km after trigonometric calculations using the cosine of a 30° angle in a right-angled triangle formed by these villages.
Step-by-step explanation:
The question is asking for the calculation of the distance from village Y to village Z, given that Y is on a bearing of 060° from X and Z is directly east of X. Since Z is directly east of X, this forms a right-angled triangle with XZ as the base, XY as the hypotenuse, and YZ as the perpendicular height. We can use trigonometry to solve this problem.
To calculate the distance YZ, we will consider the angle at X which is 060°. Since this angle is measured from North, and we need the angle from the East, we subtract it from 90° to get a 30° angle for our calculations.
We can use the cosine function for this right-angled triangle because the cosine of an angle in a right triangle is equal to the adjacent side (XZ) divided by the hypotenuse (XY).
The formula is:
cos(30°) = XZ / XY
Arranging this to solve for XZ, we get:
XZ = XY × cos(30°)
Since XY = 20 km, and cos(30°) = √3 / 2,
XZ = 20 × (√3 / 2)
XZ equals approximately 17.3 km.