An aeroplane flew from city G to city H on a bearing of 24degree. The distance G and H is 250km. It then flew a distance of 180km to city J on a bearing of 055degree. calculate (a) the distance from G to J (b) how far north of H is J
No, the plane flew on a heading.
The bearing of H from G is 24°, so that's why they set the plane's heading to 24°
But enough of that. Draw a diagram. ∡GHJ = 149°
(a) using the law of cosines, we get the distance d is
d^2 = 250^2 + 180^2 - 2*250*180cos149°
d = 414.78km
(b) Now we need to extract the x- and y-displacements from the polar forms.
If we set G at (0,0) then J is at (249.1,331.6)
All angles are measured CW from +y-axis.
a. d = GH + HJ = 250km[24o] + 180km[55o].
X = 250*sin24 + 180*sin55 = 249.1 km.
Y = 250*Cos24 + 180*Cos55 = 331.6 km.
d = 249.1 + 331.6i = 414.7km[36.9o].
b. Y1 - Y2 = 250*Cos24 - 180*Cos55 = 228.4 - 103.2 = 125.2 km.
My calculation shows H 125km higher than J.
To solve this problem, we can use trigonometry and vector components. Let's break down the steps one by one:
Step 1: Calculate the horizontal and vertical components of the airplane's movement from G to H.
To do this, we use the cosine and sine functions:
Horizontal component (GHx) = distance GH * cos(bearing) = 250 km * cos(24°)
Vertical component (GHy) = distance GH * sin(bearing) = 250 km * sin(24°)
Step 2: Calculate the ending coordinates of the airplane after flying from G to H.
The ending coordinates can be found by adding the components to the starting coordinates (which we'll assume as (0,0)):
Ending Point (Hx, Hy) = (GHx, GHy)
Step 3: Calculate the horizontal and vertical components of the airplane's movement from H to J.
We'll use the same method as in Step 1 for this calculation:
Horizontal component (HJx) = distance HJ * cos(bearing) = 180 km * cos(55°)
Vertical component (HJy) = distance HJ * sin(bearing) = 180 km * sin(55°)
Step 4: Calculate the ending coordinates of the airplane after flying from H to J.
Again, we'll add the components to the starting coordinates (which is (Hx, Hy)):
Ending Point (Jx, Jy) = (Hx + HJx, Hy + HJy)
Now let's perform the calculations:
Step 1:
GHx = 250 km * cos(24°)
GHy = 250 km * sin(24°)
Step 2:
Hx = GHx
Hy = GHy
Step 3:
HJx = 180 km * cos(55°)
HJy = 180 km * sin(55°)
Step 4:
Jx = Hx + HJx
Jy = Hy + HJy
(a) The distance from G to J:
This can be calculated using the Pythagorean theorem:
Distance from G to J = √((Jx - 0)^2 + (Jy - 0)^2)
(b) How far north of H is J:
Jy - Hy
Let's perform the calculations now.
To solve this problem, we can use trigonometry and vector components. Let's break down the given information and calculate the solutions step by step.
Given:
- Distance between city G and H = 250 km
- Bearing of the flight from G to H = 24 degrees
- Distance between city H and J = 180 km
- Bearing of the flight from H to J = 55 degrees
(a) Finding the distance from G to J:
Step 1: Determine the horizontal and vertical components of the displacement between G and H.
- The horizontal component (G to H) = 250 km * cos(24 degrees)
- The vertical component (G to H) = 250 km * sin(24 degrees)
Step 2: Determine the horizontal and vertical components of the displacement between H and J.
- The horizontal component (H to J) = 180 km * cos(55 degrees)
- The vertical component (H to J) = 180 km * sin(55 degrees)
Step 3: Add the horizontal and vertical components separately to find the total displacement from G to J.
- Horizontal displacement (G to J) = (G to H) horizontal component + (H to J) horizontal component
- Vertical displacement (G to J) = (G to H) vertical component + (H to J) vertical component
Step 4: Calculate the distance from G to J using the Pythagorean theorem.
- Distance from G to J = sqrt((Horizontal displacement)^2 + (Vertical displacement)^2)
(b) Finding how far north of H is J:
Step 1: Calculate the vertical displacement from H to J.
- Vertical displacement (H to J) = (H to J) vertical component
Step 2: Since the vertical displacement represents the distance in the north direction, we can conclude that J is located north of H.
Now, let's calculate the solutions:
(a) Calculating the distance from G to J:
- Horizontal component (G to H) = 250 km * cos(24 degrees) = 226.404 km
- Vertical component (G to H) = 250 km * sin(24 degrees) = 103.815 km
- Horizontal component (H to J) = 180 km * cos(55 degrees) = 97.134 km
- Vertical component (H to J) = 180 km * sin(55 degrees) = 144.772 km
- Horizontal displacement (G to J) = (226.404 km) + (97.134 km) = 323.538 km
- Vertical displacement (G to J) = (103.815 km) + (144.772 km) = 248.587 km
- Distance from G to J = sqrt((323.538 km)^2 + (248.587 km)^2) = 406.01 km
(b) Calculating how far north of H is J:
- Vertical displacement (H to J) = 180 km * sin(55 degrees) = 144.772 km
Therefore, J is located 144.772 km north of H.