To solve this problem, we can use the Law of Cosines, which is helpful in finding the third side of a triangle when we know two sides and the included angle. However, the problem doesn't provide an angle directly, so we will assume that the surveyor measured the distance from two points along the shoreline and that the points where the measurements are taken form a right triangle. This is a common assumption in these types of surveying problems.

Given:

• $a = 2.52$ miles
• $b = 3.46$ miles

If we assume the angle between sides $a$ and $b$ is 90 degrees, then the distance across the lake $c$ can be found using the Pythagorean theorem:

$c = \sqrt{a^2 + b^2}$

Let's calculate it:

$a^2 = (2.52)^2 = 6.3504$ $b^2 = (3.46)^2 = 11.9716$ $c = \sqrt{6.3504 + 11.9716} = \sqrt{18.322}$ $c \approx 4.28 \text{ miles}$

Thus, the distance across the lake is approximately $4.28$ miles.

Do you want any details or have any questions? Here are some related questions you might find interesting:

1. How would the problem change if the angle between $a$ and $b$ was not 90 degrees?
2. How would you find the distance if only one measurement and an angle were given?
3. What other real-world applications might use the Law of Cosines or the Pythagorean theorem?
4. How can trigonometric functions assist in solving real-world surveying problems?
5. How would you measure the distances if you had to perform the survey yourself?

Tip: In surveying and real-world applications, always verify the assumptions made (like assuming a right angle) and consider using different methods to ensure accuracy.