An+aeroplane+flies+from+a+town+A+to+a+town+B++50km+away+on+a+bearing+of+210+degree+.if+then+flew+from+town+B+to+another+town+C+on+a+bearing+of+150degree.if town C is 80km away from B,find the distance between A and C to the nearest km.find the bearing A from C to the nearest degree.find how far east B is C.
Let vector u point from A to B, and let v point from B to C. Then
u = (-50 sin30°, -50 cos30°) = (-25.00, -43.30)
v = (80 sin30°, -80 cos30°) = (40, -69.28)
Use law of cosines to find AC:
AC^2 = 50^2 + 80^2 - 2*50*80 cos120°
The vector from A to C is u+v = (15,-112.58)
bearing of A from C = 90-θ where tanθ = -15/112.58
eastward from B to C is just 40
To find the distance between town A and town C, we can use the cosine rule.
Step 1: Find the distance between town A and town B:
Using the cosine rule, we have:
AB^2 = AC^2 + BC^2 - 2 * AC * BC * cos(angle ABC)
AC = 50 km (given)
BC = 80 km (given)
angle ABC = 360 - (210 + 150) = 360 - 360 = 0 degrees
Substituting the values into the cosine rule:
AB^2 = 50^2 + 80^2 - 2 * 50 * 80 * cos(0)
AB^2 = 2500 + 6400 - 8000
AB^2 = 5400
Taking the square root of both sides:
AB = √5400
AB ≈ 73.48 km (to the nearest km)
Step 2: Find the bearing from town A to town C:
The bearing from town A to town C is the sum of the bearings from A to B and from B to C.
Bearing from A to B = 210 degrees (given)
Bearing from B to C = 150 degrees (given)
Bearing from A to C = 210 + 150 = 360 degrees
Since the bearing is a full circle, 360 degrees is the same as 0 degrees.
So, the bearing from A to C is 0 degrees.
Step 3: Find how far east B is from C:
Since the bearings are both less than 180 degrees, we can assume that A, B, and C are in a straight line.
The distance east from B to C is given by the equation:
distance east = BC * sin(angle BCA)
BC = 80 km (given)
angle BCA = 180 - 150 = 30 degrees
Substituting the values:
distance east = 80 * sin(30)
distance east = 40 km
Therefore, the distance between town A and town C is approximately 73 km.
The bearing from A to C is 0 degrees.
And, town C is located 40 km east of town B.
To solve this problem, we'll break it down into multiple steps:
Step 1: Find the distance between A and C:
To find the distance between A and C, we can use the concept of vector addition. Since we have the distances (50km and 80km) and bearings (210° and 150°), we can break each leg of the journey into its x and y components and then add them together.
First, let's find the x and y components of the distance between A and B:
The x-component of the distance from A to B is given by the formula: distance * sin(bearing)
The y-component of the distance from A to B is given by the formula: distance * cos(bearing)
Using the formula, we can calculate the x and y components for the A to B leg:
x_component_A_B = 50km * sin(210°)
y_component_A_B = 50km * cos(210°)
Next, let's find the x and y components of the distance between B and C:
Similarly, using the formulas, we can calculate the x and y components for the B to C leg:
x_component_B_C = 80km * sin(150°)
y_component_B_C = 80km * cos(150°)
Step 2: Calculate the total x and y components:
To find the x and y components of the total distance between A and C, we add the respective x and y components from step 1:
x_component_A_C = x_component_A_B + x_component_B_C
y_component_A_C = y_component_A_B + y_component_B_C
Step 3: Find the distance between A and C:
Using the x and y components from step 2, we can find the distance using the Pythagorean theorem:
distance_A_C = sqrt(x_component_A_C^2 + y_component_A_C^2)
Step 4: Round the distance between A and C to the nearest kilometer:
Round the calculated distance_A_C to the nearest kilometer to find the distance between A and C.
Step 5: Find the bearing of A from C:
The bearing of A from C can be found using the arctan function:
bearing_A_C = atan2(x_component_A_C, y_component_A_C)
Step 6: Round the bearing A from C to the nearest degree:
Round the calculated bearing_A_C to the nearest degree.
Step 7: Find how far east B is from C:
The easting distance is represented by the x-component of the B to C leg:
east_distance_B_C = x_component_B_C
Below are the calculations for each step:
Step 1: Find the x and y components for the A to B leg:
x_component_A_B = 50km * sin(210°)
y_component_A_B = 50km * cos(210°)
x_component_A_B ≈ -29.29km
y_component_A_B ≈ -40.39km
Step 2: Find the x and y components for the B to C leg:
x_component_B_C = 80km * sin(150°)
y_component_B_C = 80km * cos(150°)
x_component_B_C ≈ 68.02km
y_component_B_C ≈ -75.9km
Step 3: Calculate the total x and y components:
x_component_A_C = x_component_A_B + x_component_B_C ≈ -29.29km + 68.02km ≈ 38.73km
y_component_A_C = y_component_A_B + y_component_B_C ≈ -40.39km + (-75.9km) ≈ -116.29km
Step 4: Find the distance between A and C:
distance_A_C = sqrt(x_component_A_C^2 + y_component_A_C^2) ≈ sqrt((38.73km)^2 + (-116.29km)^2) ≈ sqrt(1499.91km^2) ≈ 38.73km
Step 5: Round the distance between A and C to the nearest kilometer:
The distance between A and C is approximately 39km.
Step 6: Find the bearing of A from C:
bearing_A_C = atan2(x_component_A_C, y_component_A_C) ≈ atan2(38.73km, -116.29km) ≈ -17.07°
Step 7: Find how far east B is from C:
east_distance_B_C = x_component_B_C ≈ 68.02km
So, to summarize the answers to the questions you asked:
- The distance between A and C is approximately 39 kilometers.
- The bearing of A from C is approximately -17 degrees.
- B is approximately 68 kilometers east of C.