An aeroplane flew from city G to city H on a bearing of 145 degree. The distance between G and H is 280 km, it then flew a distance of 430 km to city J on a bearing of 60 degree. Calculate: (a) The distance of G to J (b) How far North of H is J (c) How far west of H is G
In triangle GHJ,
∠H = 95°, so
(a) the distance GJ = side h, so
h^2 = 280^2 + 430^2 - 2*280*430 cos95°
If G is at (0,0) then
H is at (160.6, -229.4)
J is at (533.0, -14.36)
now finish it off
Given: GH = 280km[145o], HJ = 430km[60o].
a. GJ = 260km[145o]+430km[60o]
GJ = (260*sin145+430*sin60)+(260*c0s145+430*cos60)i
GJ = 522+2i = 522[0o] CW from +y-axis.
b. 360-145 = 215o.
c. 270-145 = 125o.
a. Correction: GJ = 522km[90o].
b. 145-90 = 55o.
c.
(a) The distance from G to J can be calculated using the cosine rule. Let's call the distance from H to J x:
x^2 = 280^2 + 430^2 - 2 * 280 * 430 * cos(145 - 60)
Solving this equation will give us the answer to (a).
(b) To find out how far north of H city J is, we need to calculate the vertical component of the distance from H to J. This can be done by multiplying the distance from H to J (found in part (a)) by the sine of the angle between the two cities.
(c) To determine how far west of H city G is, we can multiply the distance from G to J (again found in part (a)) by the cosine of the bearing from J to H.
To solve this problem, we will use trigonometry and vector addition.
(a) Distance from G to J:
First, let's break down the journey from G to J into two components:
1. Distance traveled in the northward direction from H to J.
To calculate this distance, we can use trigonometry and the given bearing of 60 degrees. Since the bearing is measured clockwise from the north, we can calculate the northward component using the sine function:
Distance north (HN) = 430 km * sin(°60) = 430 km * 0.866 = 372.38 km (Approximately)
2. Distance traveled in the eastward direction from H to J.
To calculate this distance, we can use trigonometry and the given bearing of 60 degrees. Since the bearing is measured clockwise from the north, we can calculate the eastward component using the cosine function:
Distance east (EJ) = 430 km * cos(°60) = 430 km * 0.5 = 215 km
Now, we can calculate the total distance from G to J using the Pythagorean theorem:
Distance (GJ) = √(HN² + EJ²) = √(372.38² + 215²) ≈ 432.23 km
Therefore, the distance from G to J is approximately 432.23 km.
(b) How far north of H is J:
We already found the northward component of the distance from H to J, which is HN = 372.38 km. Therefore, J is approximately 372.38 km north of H.
(c) How far west of H is G:
To find the westward component, we need to break down the distance from G to H into northward and westward components.
1. Distance traveled in the northward direction from G to H.
Since we are given the bearing of 145 degrees, we can calculate the northward component using the sine function:
Distance north (GH) = 280 km * sin(°145) = 280 km * (-0.819) ≈ -229.22 km (Approximately)
Note: The negative sign indicates a southward direction.
2. Distance traveled in the westward direction from G to H.
To calculate this distance, we can use trigonometry and the given bearing of 145 degrees. We will use the cosine function since the bearing is measured clockwise from the north:
Distance west (WG) = 280 km * cos(°145) = 280 km * (-0.573) ≈ -162.84 km (Approximately)
Note: The negative sign indicates a westward direction.
Therefore, G is approximately 229.22 km south and 162.84 km west of H.