- Identify the given parameters: distance traveled by each cyclist \(5\) miles and the angle between their paths \(30^\circ\).
- Use the Law of Cosines \(d^2 = a^2 + b^2 - 2ab \cos(\theta)\) to relate the distances and the angle.
- Calculate the distance \(d = \sqrt{5^2 + 5^2 - 2(5)(5) \cos(30^\circ)}\).
- Simplify the expression to find the final distance, which is \(\boxed{5\sqrt{2-\sqrt{3}}}\) miles.
Explanation
- Define the problem and parameters.
We are given that two cyclists start from the same point \(O\) and travel along paths that meet at an angle of \(30^\circ\). Each cyclist travels a distance of \(5\) miles. Let \(A\) be the final position of the first cyclist and \(B\) be the final position of the second cyclist. Therefore, the distance from the starting point to each cyclist is \(OA = 5\) and \(OB = 5\). The angle between these two paths is \(\angle AOB = 30^\circ\). We need to find the distance between the two cyclists, which corresponds to the length of the segment \(AB\), denoted as \(d\).
- Apply the Law of Cosines.
To find the distance \(d\) between points \(A\) and \(B\) in triangle \(OAB\), we apply the Law of Cosines. The Law of Cosines states that for any triangle with sides \(a\), \(b\), and \(c\) and an angle \(\theta\) opposite side \(c\):
\[c^2 = a^2 + b^2 - 2ab \cos(\theta)\]
In our scenario, \(a = OA = 5\), \(b = OB = 5\), the included angle is \(\theta = 30^\circ\), and the side we want to find is \(c = d = |AB|\). Substituting these values into the formula:
\[d^2 = 5^2 + 5^2 - 2(5)(5) \cos(30^\circ)\]
- Perform calculations and determine the final result.
First, calculate the squares and the product:
\[d^2 = 25 + 25 - 50 \cos(30^\circ)\]
\[d^2 = 50 - 50 \cos(30^\circ)\]
Next, substitute the value of \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\):
\[d^2 = 50 - 50 \left( \frac{\sqrt{3}}{2} \right)\]
\[d^2 = 50 - 25\sqrt{3}\]
To find \(d\), take the square root of both sides:
\[d = \sqrt{50 - 25\sqrt{3}}\]
Factoring out \(25\) from under the square root:
\[d = \sqrt{25(2 - \sqrt{3})} = 5\sqrt{2 - \sqrt{3}}\]
The distance between the two cyclists is \(5\sqrt{2 - \sqrt{3}}\) miles.
Examples
The Law of Cosines is incredibly useful in real-world navigation and surveying. For instance, if you are a pilot or a sailor tracking the movement of two ships leaving a port at different angles, you can use the Law of Cosines to calculate the distance between them at any given time. This calculation is essential for maintaining safety, planning routes, and predicting future positions based on speed and heading. By simply knowing the distance traveled by each vessel and the angle between their paths, you can instantly determine their separation distance without needing to physically measure it.