Boat's Journey Calculating The Change In Direction

This mathematical problem presents a fascinating scenario involving a boat's journey, challenging us to determine the change in direction using geometric principles and the Law of Cosines. The boat initially travels north for 28 miles, then alters its course by an unknown angle, $x^{\circ}$, towards the southwest, continuing for 25 miles before halting. The final position of the boat is 18 miles away from its starting point. Our mission is to calculate the value of $x$, representing the change in the boat's direction. This problem is not just a mathematical exercise; it's a real-world application of trigonometry and geometry, demonstrating how these concepts can be used to solve navigation and spatial reasoning problems. Understanding the Law of Cosines is paramount to solving this problem, as it allows us to relate the sides and angles of a triangle, even when it's not a right triangle. Additionally, visualizing the boat's path as a triangle is crucial. The sides of the triangle represent the distances traveled, and the angle opposite the side connecting the starting and ending points is key to finding the change in direction. The challenge lies in correctly applying the Law of Cosines and interpreting the result in the context of the boat's changing direction. This requires careful attention to detail and a solid understanding of trigonometric relationships. Let's embark on this journey of mathematical discovery and unravel the mystery of the boat's change in direction.

To effectively solve this problem, a visual representation is indispensable. Imagine the boat's journey as a triangle. The first leg of the journey, 28 miles north, forms one side of the triangle. The second leg, 25 miles southwest, forms another side. The direct distance from the starting point to the stopping point, 18 miles, closes the triangle, forming the third side. Now, let's delve into the heart of the problem: the angle. The boat initially traveled north, and then turned $x^{\circ}$ southwest. This turn creates an interior angle within our triangle. However, the angle we need for the Law of Cosines is the angle opposite the 18-mile side. This angle is not simply $x^{\circ}$. We need to consider that southwest is 45 degrees from both south and west. Therefore, the interior angle of the triangle is related to $x^{\circ}$, but it's not a direct equivalent. Let's call the interior angle of the triangle opposite the 18-mile side $\theta$. We need to find a relationship between $x$ and $\theta$. Visualizing the directions and the turn, we can see that the angle between the northward path and the southwestward path is $180^{\circ} - x^{\circ}$. However, since southwest is 45 degrees from south, the interior angle $\theta$ can be expressed as $180^{\circ} - (180^{\circ} - x^{\circ} + 45^{\circ})$, which simplifies to $x^{\circ} - 45^{\circ}$. This is a critical step in setting up the problem correctly. We now have a triangle with sides of 28 miles, 25 miles, and 18 miles, and the angle opposite the 18-mile side is related to the unknown change in direction, $x$. With this clear geometric representation and the relationship between $x$ and $\theta$ established, we are well-prepared to apply the Law of Cosines and solve for the unknown.

The Law of Cosines is a fundamental trigonometric principle that allows us to relate the lengths of the sides of a triangle to the cosine of one of its angles. In our boat journey problem, we have a triangle with sides of lengths 28 miles, 25 miles, and 18 miles. Let's denote these sides as $a = 28$, $b = 25$, and $c = 18$. The angle opposite the side of length $c$ (18 miles) is the angle $\theta$, which, as we established earlier, is related to the change in direction $x$ by the equation $\theta = x - 45^{\circ}$. The Law of Cosines states: $c^2 = a^2 + b^2 - 2ab \cos(\theta)$. Now, let's substitute the known values into the equation: $18^2 = 28^2 + 25^2 - 2(28)(25) \cos(\theta)$. This equation forms the cornerstone of our solution. We have one equation with one unknown, $\cos(\theta)$. Our next step is to isolate $\cos(\theta)$ and solve for its value. First, calculate the squares: $324 = 784 + 625 - 1400 \cos(\theta)$. Then, simplify and rearrange the equation to isolate the term with $\cos(\theta)$: $1400 \cos(\theta) = 784 + 625 - 324$. This simplifies to: $1400 \cos(\theta) = 1085$. Now, divide both sides by 1400 to solve for $\cos(\theta)$: $\cos(\theta) = \frac{1085}{1400}$. This gives us the value of the cosine of the angle $\theta$. The next step is to find the angle $\theta$ itself by taking the inverse cosine (also known as arccosine) of this value. This will give us the measure of the interior angle of the triangle, which is crucial for determining the change in direction, $x$.

Having isolated $\cos(\theta)$ in the previous step, we now have $\cos(\theta) = \frac{1085}{1400}$. To find the angle $\theta$ itself, we need to use the inverse cosine function, often denoted as $\arccos$ or $\cos^{-1}$. This function essentially