To find the distance across the lake, we can use the Law of Cosines, provided we have all necessary information about the angles or additional side lengths. However, the problem does not specify an angle between the two given sides (a and b). Assuming that the distance across the lake is the line between points A and B, the problem likely involves a triangle, and we might need to assume a right angle, unless other information is given.

Let's denote:

• a = 2.52 miles (distance from a point to point A)
• b = 3.46 miles (distance from the same point to point B)
• c = distance across the lake (the distance between points A and B)

Case 1: If we assume a right triangle

In this case, the Pythagorean Theorem would be applicable: $c^2 = a^2 + b^2$ $c = \sqrt{a^2 + b^2}$

Substituting the values: $c = \sqrt{(2.52)^2 + (3.46)^2}$

Let me calculate this for you.The distance across the lake is approximately 4.28 miles, assuming a right triangle.

Would you like further clarification on this or any other details?

Here are five related questions you might find useful:

1. What is the Law of Cosines, and how does it differ from the Pythagorean theorem?
2. How do you calculate distances when dealing with non-right triangles?
3. How do angles affect the calculation of distances between two points?
4. What are other common applications of the Pythagorean theorem in surveying?
5. How can trigonometric functions help in solving more complex surveying problems?

Tip: Always double-check the assumptions in geometry problems, as they can drastically change the method used to solve the problem.