Cecilia is sailing on Lake Erie just west of Kelly's Island. She can see two lighthouses from her location. The lighthouse at South Bass Island is 4.6 miles away on a bearing of $\mathrm{N} 67^{\circ} \mathrm{W}$. The lighthouse at Catawba Island is 5.4 miles away on a bearing of $\mathrm{S} 27^{\circ} \mathrm{E}$. (Source, www topozone com). Determine the distance between the two lighthouses, rounded to one decimal place.

Answer: The distance between the two lighthouses is Number miles.

Step 1 ：Given that Cecilia is sailing on Lake Erie just west of Kelly's Island and can see two lighthouses from her location. The lighthouse at South Bass Island is 4.6 miles away on a bearing of \(N 67^{\circ} W\). The lighthouse at Catawba Island is 5.4 miles away on a bearing of \(S 27^{\circ} E\).

Step 2 ：We are asked to determine the distance between the two lighthouses, rounded to one decimal place.

Step 3 ：This problem can be solved using the law of cosines. The law of cosines states that for any triangle with sides of lengths a, b, and c and an angle \(γ\) opposite side c, the following equation holds: \(c² = a² + b² - 2abcos(γ)\).

Step 4 ：In this case, the sides a and b are the distances from Cecilia to the two lighthouses, and the angle \(γ\) is the difference between the bearings of the two lighthouses.

Step 5 ：We can calculate \(γ\) by converting the bearings to angles measured counterclockwise from due east, then subtracting the smaller angle from the larger one.

Step 6 ：After finding \(γ\), we can use the law of cosines to find the distance between the two lighthouses.

Step 7 ：Let's denote the distance from Cecilia to South Bass Island as \(a = 4.6\) miles, and the distance from Cecilia to Catawba Island as \(b = 5.4\) miles.

Step 8 ：The bearing of South Bass Island is \(N 67^{\circ} W\), which corresponds to an angle of \(23^{\circ}\) measured counterclockwise from due east. Similarly, the bearing of Catawba Island is \(S 27^{\circ} E\), which corresponds to an angle of \(117^{\circ}\) measured counterclockwise from due east.

Step 9 ：The angle \(γ\) between the two lighthouses is the difference between the two angles, which is \(γ = 117^{\circ} - 23^{\circ} = 94^{\circ}\).

Step 10 ：Substituting the values of \(a\), \(b\), and \(γ\) into the law of cosines, we get \(c² = 4.6² + 5.4² - 2*4.6*5.4*cos(94^{\circ})\).

Step 11 ：Solving for \(c\), we get \(c = \sqrt{4.6² + 5.4² - 2*4.6*5.4*cos(94^{\circ})} = 7.3\) miles.

Step 12 ：Final Answer: The distance between the two lighthouses is \(\boxed{7.3}\) miles.