An aero plane flies from a town X on a bearing of N45°E to another town Y, a distance of 200km. it then changes course and flies to another town Z on a bearing of S60°E. If Z is directly to east of X, calculate,
(a) Draw a comprehensive diagram that illustrate the information above
(b) The distance from X to Z
(c) The distance from Y to XZ
using the law of sines,
(b) XZ/sin105° = 200/sin30°
(c) 200 sin45°
To TAKE a bearing on something, like a lighthouse bears northeast.
You fly or sail or walk on a HEADING.
Math text writers do not do navigation.
Anyway
how far east of x is y? 200 sin 45 call that a = 141.4
how far east of y is z?
well y is 200 cos 45 = 141.4 north of x (that is part c by the way)
so we must go 141.4 south to get to y
that means tan 60 = b/141.4 = 1.732
so b (east distance yz) = 244.9
a+b = x to z east = 386.3 ( part b)
To answer the questions, we will follow the steps below:
(a) Drawing a comprehensive diagram:
First, draw a line segment XY to represent the distance from town X to Y. Use a ruler to make it 200 km long.
Next, label point Z directly to the east of point X.
To find the bearing from X to Y, measure an angle of 45° in the clockwise direction from the north (upwards direction) and draw a line segment from X to the point where the angle intersects the line XY. Label this point as A.
To find the bearing from X to Z, measure an angle of 60° in the clockwise direction from the south (downwards direction) and draw a line segment from X to the point where the angle intersects the line XZ. Label this point as B.
Finally, draw line segments AB and AZ to complete the diagram.
(b) Calculating the distance from X to Z:
We know that angle XAB is 90°, and angle XBA is 60°. Therefore, angle ABX is 180° - 90° - 60° = 30°.
Using trigonometry, we can use the sine function to find the length of XZ. The formula is as follows:
sin(angle) = opposite / hypotenuse
In this case, sin(30°) = XZ / 200 km
Rearranging the formula, we get:
XZ = 200 km * sin(30°)
Using a calculator, evaluate sin(30°) ≈ 0.5
XZ = 200 km * 0.5 = 100 km
Therefore, the distance from X to Z is 100 km.
(c) Calculating the distance from Y to XZ:
We can use the Pythagorean theorem to find the length of YXZ.
In right triangle XYZ, with XY = 200 km and XZ = 100 km, we need to find YX.
Using the Pythagorean theorem:
YX² = XY² - XZ²
YX² = (200 km)² - (100 km)²
YX² = 40000 km² - 10000 km²
YX² = 30000 km²
Taking the square root of both sides:
YX = √(30000 km²)
YX ≈ 173.21 km
Therefore, the distance from Y to XZ is approximately 173.21 km.