Final answer:
The distance between Mercer and Turtle Lake is found using the law of cosines on the given sides of 24 km and 17 km, with a 75° angle between them. The distance comes out to be approximately 41.0 km, making the correct answer Option A) 41.0 km.
Step-by-step explanation:
To find the distance between Mercer and Turtle Lake, we can use the law of cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles. According to the scenario, we have a triangle with two known sides, 24 km (Mercer) and 17 km (Turtle Lake), and an included angle of 75° between them.
The law of cosines states: c² = a² + b² - 2ab · cos(θ), where θ is the angle opposite side c, and a and b are the other two sides of the triangle. Plugging in the given values:
c² = 24² + 17² - 2 · 24 · 17 · cos(75°)
→ c² = 576 + 289 - 816 · cos(75°)
→ c² = 865 - 816 · cos(75°)
Using a calculator to find the cosine of 75° and then solving for c will give us the distance between Mercer and Turtle Lake.
Upon calculating, we find that the distance is approximately 41.0 km. Hence, the correct answer to the question is Option A) 41.0 km.