An airplane is flying at a bearing of 142 degrees at 275 mph. The wind affecting the plane is blowing at a bearing of 280 degrees at 35 mph. Determine the actual bearing of the airplane. Round your answer to the nearest hundredth and do not include the units.
you take a bearing on a distant object
you fly on a heading.
Just adding the two vectors, you get
275cis142° + 35cis280° = 250.05 @ 147.25°
That is, 250 mph on a heading of 147°
To determine the actual bearing of the airplane, we need to consider the effect of the wind on the airplane's direction.
1. Start by drawing a diagram to visualize the problem. Draw a horizontal line to represent the ground, and draw a line pointing to the east to represent the airplane's initial bearing of 142 degrees. Label this line "Airplane."
2. Now, we need to consider the effect of the wind. Draw another line starting from the end of the "Airplane" line, pointing in the direction of the wind's bearing of 280 degrees. Label this line "Wind."
3. Next, we need to find the resultant vector, which represents the actual direction of the airplane. To do this, we can use vector addition.
- First, break down the vectors into their components. The "Airplane" vector has an eastward component of (275 mph) * cos(142 degrees) and a northward component of (275 mph) * sin(142 degrees).
- Similarly, the "Wind" vector has an eastward component of (35 mph) * cos(280 degrees) and a northward component of (35 mph) * sin(280 degrees).
4. Add the eastward components and the northward components separately to find the resultant vector's components.
- Eastward component: (275 mph) * cos(142 degrees) + (35 mph) * cos(280 degrees)
- Northward component: (275 mph) * sin(142 degrees) + (35 mph) * sin(280 degrees)
5. The resultant vector's components give us the eastward and northward displacements of the airplane caused by both its original bearing and the wind.
6. Now, we can find the actual bearing of the airplane using the tangent function.
- Actual bearing = arctan(northward component / eastward component)
7. Plug in the values from Step 4 into the formula to calculate the actual bearing.
8. Round the answer to the nearest hundredth and exclude the units to obtain the final result.
By following these steps, you can accurately determine the actual bearing of the airplane given the provided information.