If an airplane flies 100 miles due north, then turns and flies 100 miles directly northeast, how far is it then from the original starting point, to the nearest mile? margin 3 miles for calculation errors.

When an aircraft embarks on a journey, its total travel distance is simply the sum of all the segments flown. However, its displacement, or the direct distance from its starting point to its final position, can be quite different. This is a classic problem in vector addition, where the direction of each segment significantly impacts the final outcome.

To determine how far the plane is from its origin, we can visualize its path as two sides of a triangle. The first leg of the journey, 100 miles due north, forms one side. The second leg, 100 miles directly northeast, forms another. The key to solving this lies in understanding the angle between these two segments when they meet. If the plane flies north and then turns to fly northeast, the angle of the turn is 45 degrees. Therefore, the internal angle of the triangle formed by the starting point, the turning point, and the final position is 180 degrees minus this 45-degree turn, which equals 135 degrees.

With two sides of the triangle known (100 miles each) and the angle between them (135 degrees), we can employ a powerful tool from trigonometry: the Law of Cosines. This mathematical rule allows us to find the length of the third side of a triangle when we know two sides and the included angle. The formula essentially states that the square of the unknown side is equal to the sum of the squares of the other two sides, minus twice the product of those two sides multiplied by the cosine of the angle between them.

Applying the Law of Cosines, with the two 100-mile segments and the 135-degree angle, we find that the square of the direct distance is approximately 34142 square miles. Taking the square root of this value yields approximately 184.77 miles. Rounded to the nearest mile, the airplane is 185 miles from its original starting point. This type of calculation is fundamental in real-world navigation, enabling pilots and navigators to determine precise positions and optimal routes.