An aeroplane flies from a town X on a bearing of N 45 degree E to another a town Y on a distance of 200km.it then changes course and flies to another town Z on a bearing of S 60 degree E.if Z is directly east of X.Calculate (a) the distance from X to Y (b) the distance from Y to XZ
Answer
show the diagram.
heading, not bearing.
From Y, Z is on a bearing of S60E, so that is the plane's heading.
200 @ N45E = <141.42,141.42>
z @ S60E = <.866z,-.50z>
Since Z is directly east of X, you need
141.42 - 0.50z = 0
So, evaluate the distance z, and then the distance XZ will be 141.42 + 0.866z
Show the diagram
Show the diagram
To solve this problem, we can break it down into smaller steps using trigonometry and geometry.
First, let's calculate the distance from X to Y:
Step 1: Draw a diagram.
Draw a diagram that represents the given scenario. X is the starting point, Y is the destination after the first leg of the flight, and Z is the final destination.
Step 2: Determine the angles of flight.
The plane flies on a bearing of N 45° E from X to Y, which means it travels northeast. This angle forms a right-angled triangle with the ground and the hypotenuse is the distance from X to Y.
Step 3: Use trigonometry to find the distance.
We can use trigonometry, specifically the sine and cosine functions, to find the distance from X to Y. The sine of an angle is the opposite side divided by the hypotenuse, and the cosine of an angle is the adjacent side divided by the hypotenuse.
In this case, since we know the angle and the hypotenuse (200 km), we can use the sine function to find the opposite side (distance from X to Y).
sin(45°) = opposite / hypotenuse
sin(45°) = xy / 200
Rearranging the equation:
xy = sin(45°) * 200
Calculating the value:
xy = 0.7071 * 200
xy = 141.42 km
So, the distance from X to Y is 141.42 km.
Now, let's calculate the distance from Y to XZ:
Step 4: Determine the angles of flight.
The plane flies on a bearing of S 60° E from Y to Z, which means it travels southeast. This angle forms another right-angled triangle with the ground, and we need to find the hypotenuse, which is the distance from Y to Z, along with the adjacent side, which is the distance from X to Z.
Step 5: Use trigonometry to find the distance.
Since we have the adjacent side (distance from X to Z) and the angle, we can use the cosine function to find the hypotenuse (distance from Y to Z).
cos(60°) = adjacent / hypotenuse
cos(60°) = xz / hy
Rearranging the equation:
xz = cos(60°) * hy
We know that hy equals xy (distance from X to Y), which we had calculated earlier.
xz = cos(60°) * 141.42
Calculating the value:
xz = 0.5 * 141.42
xz = 70.71 km
So, the distance from Y to XZ is 70.71 km.
To summarize:
(a) The distance from X to Y is 141.42 km.
(b) The distance from Y to XZ is 70.71 km.