need help with this work question

Question

To find the distance across a small lake, a surveyor has taken the measurements shown. Find the distance across the lake using this information. (Assume points A and B are exactly along the shoreline, and that a = 2.82 miles and b = 3.11 miles. Round your answer to two decimal places.)

 mi

Answer 1

We shall use the Law of Cosines.

The Law of Cosines states that given a triangle ABC, where A, B, C are the angles opposite side lengths a, b, c, c² = a² + b² - 2(a)(b)cos(C).

In this problem, we want to find the distance across the lake from point A to B. This is really asking us to find the length of side c.

We know that a = 2.82, b = 3.11, and C = 40.3, as given in the problem. Plugging these values into the formula, we have

c² = (2.82)² + (3.11)² - 2(2.82)(3.11)cos(40.3)

c² = 7.9524 + 9.6721 - 17.5404(0.76266)

c² = 4.24699243

c = 2.060823241 ≈ 2.06

Thus, our answer is approximately 2.06 miles.

Answer 2

by cosine law AB = (a² + b² - 2abcosC)^1/2 = (2.82² + 3.11² - 2·2.82·3.11·cos40.3°)^1/2 ≈ 2.06 mi