an airplane leaves the airport and flies due west 150 miles and then 240 miles in the direction 200°40'. Assuming the earth is flat, how far is the plane from the airport at this time( to the nearest mile)?
I will assume that you are using the standard aircraft navigation which has 0 degrees at North and goes clockwise.
So I see a triangle with sides 150 and 240 with a contained angle of 110.666667 degrees
Using the cosine law ...
so distance^2 = 150^2 + 240^2 - 2(150)(240)cos110.66667
= 105511.0006
distance = √105511.0006
= 324.8 miles
To find the distance of the plane from the airport, we can use the concept of vector addition. We need to calculate the horizontal and vertical components of the two flight segments separately and then add them up to get the total displacement.
1. The first segment is due west for 150 miles. Since the plane is flying directly west, there is no vertical displacement, only a horizontal one. Therefore, the horizontal (x-component) displacement is 150 miles.
2. The second segment is given as 240 miles in the direction 200°40'. To find the horizontal and vertical components, we can use trigonometry. The horizontal component is given by: 240 * cos(200°40'). The vertical component is given by: 240 * sin(200°40').
Calculating these values:
Horizontal component = 240 * cos(200°40')
Vertical component = 240 * sin(200°40')
3. Next, we add up the horizontal components from both segments: 150 miles + horizontal component of the second segment.
4. Similarly, we add up the vertical components from both segments.
5. Finally, we can use the Pythagorean theorem to calculate the distance of the plane from the airport. The distance is the square root of the sum of the horizontal squared and vertical squared components.
Here are the steps to calculate the distance:
Step 1: Calculate the horizontal displacement of the second segment:
horizontal component = 240 * cos(200°40')
Step 2: Calculate the vertical displacement of the second segment:
vertical component = 240 * sin(200°40')
Step 3: Add up the horizontal components from both segments:
total horizontal displacement = 150 miles + horizontal component
Step 4: Add up the vertical components from both segments:
total vertical displacement = vertical component
Step 5: Calculate the distance from the airport:
distance = sqrt((total horizontal displacement)^2 + (total vertical displacement)^2)
Now you can substitute the values into the formulas and use a calculator to compute the final result, rounding to the nearest mile.