a plane flies 810 miles form Franklin to Centerville with a bearing of 75 degrees (clockwise from north). then it flies 648 miles from centerville to rosemont with a bearing of 32 degrees. find the straight-line distance and bearing from rosemont to franklin.
Compute the north and east displacements of both legs of the journey.
Add the two north displacements. Call that sum Y
Y = 810 cos75 + 648 cos32 = __
Add the two east displacements. Call that sum X.
X = 810 sin75 + 648 sin32 = ___
Straight-line distance = sqrt[X^2 + Y^2]
To find the straight-line distance and bearing from Rosemont to Franklin, we can use vector addition.
Let's start by labeling the points:
Franklin is point A,
Centerville is point B,
Rosemont is point C.
Given information:
Distance from Franklin to Centerville (AB) = 810 miles,
Bearing of AB from north = 75 degrees clockwise.
Distance from Centerville to Rosemont (BC) = 648 miles,
Bearing of BC from north = 32 degrees clockwise.
Step 1: Calculate the coordinates of points A, B, and C.
Let's assume the origin (0,0) is at a convenient location.
Given the bearing, we can break each leg of the journey into its x and y components by using trigonometry.
AB (Franklin to Centerville):
ABx = 810 * sin(75)
ABy = 810 * cos(75)
BC (Centerville to Rosemont):
BCx = 648 * sin(32)
BCy = 648 * cos(32)
Step 2: Calculate the coordinates of point C (Rosemont).
Cx = BCx + Bx
Cy = BCy + By
Step 3: Calculate the straight-line distance from Rosemont to Franklin (AC).
AC = √((Cx - Ax)² + (Cy - Ay)²)
Step 4: Calculate the bearing of AC from north.
tan(theta) = (Cx - Ax) / (Cy - Ay)
AC bearing = arctan(tan(theta))
Let's plug in the numbers:
ABx = 810 * sin(75) = 789.2
ABy = 810 * cos(75) = 191.2
BCx = 648 * sin(32) = 344.6
BCy = 648 * cos(32) = 547.6
Bx = ABx = 789.2
By = ABy = 191.2
Cx = BCx + Bx = 344.6 + 789.2 = 1133.8
Cy = BCy + By = 547.6 + 191.2 = 738.8
AC = √((Cx - Ax)² + (Cy - Ay)²) = √((1133.8 - 0)² + (738.8 - 0)²) = √(1133.8² + 738.8²) ≈ 1349.8 miles
tan(theta) = (Cx - Ax) / (Cy - Ay) = 1133.8 / 738.8 ≈ 1.534
AC bearing ≈ arctan(1.534) ≈ 56.7 degrees (clockwise from north)
Therefore, the straight-line distance from Rosemont to Franklin is approximately 1349.8 miles, and the bearing is approximately 56.7 degrees (clockwise from north).
To find the straight-line distance and bearing from Rosemont to Franklin, we can use the concept of vector addition.
Step 1: Convert the given information into vector form.
The plane traveled 810 miles from Franklin to Centerville at a bearing of 75 degrees. This can be represented as a vector with a magnitude of 810 and a direction of 75 degrees.
To convert the bearing into Cartesian coordinates (x, y), we use the following formulas:
x = magnitude * cos(bearing)
y = magnitude * sin(bearing)
For the vector from Franklin to Centerville, the x-component would be:
x1 = 810 * cos(75)
And the y-component would be:
y1 = 810 * sin(75)
Similarly, for the vector from Centerville to Rosemont, we have:
x2 = 648 * cos(32)
y2 = 648 * sin(32)
Step 2: Find the total displacement vector.
To find the total displacement vector from Franklin to Rosemont, we need to add the individual vectors together:
x_total = x1 + x2
y_total = y1 + y2
Step 3: Calculate the straight-line distance.
The straight-line distance can be found using the Pythagorean theorem:
distance = sqrt(x_total^2 + y_total^2)
Step 4: Calculate the bearing.
The bearing can be found using inverse tangent (tan^(-1)).
bearing = atan(y_total / x_total)
Plugging in the values from the previous steps into these formulas will give us the answer.
Let's calculate the values:
For the vector from Franklin to Centerville:
x1 = 810 * cos(75) = -179.61 miles
y1 = 810 * sin(75) = 773.5 miles
For the vector from Centerville to Rosemont:
x2 = 648 * cos(32) = 546.35 miles
y2 = 648 * sin(32) = 343.94 miles
Total displacement vector:
x_total = -179.61 + 546.35 = 366.74 miles
y_total = 773.5 + 343.94 = 1117.44 miles
Distance:
distance = sqrt(366.74^2 + 1117.44^2) = 1184.42 miles
Bearing:
bearing = atan(1117.44 / 366.74) = 72.15 degrees (clockwise from north)
Therefore, the straight-line distance from Rosemont to Franklin is approximately 1184.42 miles with a bearing of 72.15 degrees (clockwise from north).