A pilot flies in a straight path for 1 h 30 min. She then makes a course correction, heading 10 degrees to the right of her original course, and flies 2 h in the new direction. If she maintains a constant speed of 680 mi/h, how far is she from her starting position?
1020 miles in the first direction and 1360 miles in the second direction. Add them as vectors.
The distance traveled has components of
1360 + 1020 cos 10 = 2364.5 in the final direction and 1020 sin 10 = 177.1 in the perpendicular direction. The resultant is
sqrt (2364.5^2 + 177.1^2) = 2371.1 miles
To solve this problem, we can break it down into two parts: the distance covered in the original path and the distance covered in the new direction after the course correction.
First, let's calculate the distance covered in the original path. We know that the pilot flies for 1 hour and 30 minutes, which is equivalent to 1.5 hours. Since the pilot is maintaining a constant speed of 680 miles per hour, we can simply multiply the time by the speed to get the distance:
Distance in original path = Speed × Time
= 680 mi/h × 1.5 h
= 1020 miles
So, the pilot covers 1020 miles in the original path.
Next, let's calculate the distance covered after the course correction. We are given that the pilot flies for 2 hours in the new direction, which is 10 degrees to the right of her original course. Since the pilot is still maintaining a constant speed of 680 miles per hour, we can again multiply the time by the speed to get the distance:
Distance in new direction = Speed × Time
= 680 mi/h × 2 h
= 1360 miles
So, the pilot covers 1360 miles in the new direction after the course correction.
Now, we need to find the total distance from the starting position. To do that, we can use the Pythagorean theorem, as the pilot has traveled in a straight line from the starting position to the end of the original path and then in a straight line in the new direction after the course correction.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In this case, one side is the distance covered in the original path (1020 miles), the other side is the distance covered in the new direction after the course correction (1360 miles), and the hypotenuse is the total distance from the starting position.
Using the Pythagorean theorem, we can calculate the total distance:
Total distance = √(Distance in original path)^2 + (Distance in new direction)^2
= √(1020 miles)^2 + (1360 miles)^2
= √(1040400 miles^2 + 1849600 miles^2)
= √(2890000 miles^2)
= 1700 miles
Therefore, the pilot is approximately 1700 miles away from her starting position.