A plane flies 810 miles from A to B with a bearing of 75 degrees. Then it flies 648 miles from B to C with a bearing of 32 degrees. Find the straightline distance from and bearing from C to A.
To find the straight-line distance from C to A, we can use the Pythagorean theorem. The distances AB and BC form a right triangle with AC as the hypotenuse.
1. Calculate the length of the base, AB:
Using the cosine rule, we can find angle A (the angle opposite side AB) in the triangle ABC:
cos(A) = (b^2 + c^2 - a^2) / (2 * b * c)
cos(A) = (810^2 + 648^2 - 2 * 810 * 648 * cos(75)) / (2 * 810 * 648)
cos(A) = (656100 + 419904 - 1052442.6) / 1051200
cos(A) = 0.96765
A ≈ cos^(-1)(0.96765) ≈ 14.24 degrees
So, angle A is approximately 14.24 degrees.
Now, we have a right triangle ABC with angle A being approximately 14.24 degrees, side BC measuring 648 miles, and side AC being the hypotenuse.
2. Use the sine rule to find the length of the hypotenuse, AC (the straight-line distance from C to A):
sin(A) / AB = sin(C) / BC
sin(14.24) / AB = sin(90) / 648
AB ≈ (sin(14.24) * 648) / sin(90)
AB ≈ 238 miles
Therefore, the distance from C to A is approximately 238 miles.
3. To find the bearing from C to A, we can use the following formula:
bearing = 180 degrees + angle opposite side BC
In our case, since angle A is the angle opposite side BC, we can write:
bearing = 180 degrees + 14.24 degrees
bearing ≈ 194.24 degrees
Therefore, the bearing from C to A is approximately 194.24 degrees.
To find the straight-line distance from C to A, you can use the Pythagorean theorem because the triangle formed by points A, B, and C is a right triangle. To find the bearing from C to A, you can use trigonometry to find the angle between the line segment connecting C to A and the north direction.
Let's break down the problem step by step:
Step 1: Find the coordinates of points A, B, and C.
To solve this problem, we need the coordinates of the three points A, B, and C or at least the relative positions between them. Without this information, it is not possible to calculate the straight-line distance or the bearing accurately.
Step 2: Analyze the triangle formed by points A, B, and C.
Once we have the coordinates or relative positions between points A, B, and C, we can determine the lengths of the sides and angles of the triangle using distance formula and trigonometry calculations.
Step 3: Calculate the straight-line distance from C to A using the Pythagorean theorem.
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the straight-line distance from C to A is the hypotenuse of the triangle formed by points A, B, and C. You can use the lengths of AB and BC to find the length of AC.
Step 4: Calculate the bearing from C to A using trigonometry.
To find the bearing from C to A, you need to determine the angle between the line segment connecting C to A and the north direction. You can use the tangent of this angle to calculate its value.
In this case, you can use the coordinates or relative positions of points A, B, and C to find the slope of the line segment connecting C to A. Then, you can use the arctangent function to find the angle between the line segment and the north direction.
By following these steps, you should be able to calculate the straight-line distance from C to A and the bearing from C to A accurately.
actually, planes fly on headings, not bearings.
Setting A at (0,0), B is at
(810 sin 75, 810 cos 75) = (782,210)
moving from B to C adds another
(648 sin 32, 648 cos 32) = (343,550)
so C is at (1125,760)
distance is 1358
heading from C to A is 236