Question

A lighthouse worker is tracking a boat that is 5 km south of her. She is also tracking a boat that is 6 km from the tower located at 70 degrees east of north. To the nearest tenth of a kilometer, how far apart are the two boats?

A) 1.2 km
B) 4.2 km
C) 7.8 km
D) 11.0 km

Answer 1

By constructing a right triangle and applying the Law of Cosines with distances from the lighthouse and the angle between them, we find that the two boats are approximately 7.8 km apart, matching answer choice C.

Step-by-step explanation:

The question asks to determine the distance between two boats given their positions relative to a lighthouse and some bearings. To find the distance between the two boats, we can construct a right triangle where one boat is 5 km directly south of the lighthouse and the second boat is 6 km away in a direction 70 degrees east of north. We will use the law of cosines to find the distance between the two boats.

By using the Law of Cosines:

• Let c be the distance between the boats.
• a = 5 km is the distance of the first boat from the lighthouse.
• b = 6 km is the distance of the second boat from the lighthouse.
• Angle γ = 70° is the angle at the lighthouse between the lines to the boats.

The Law of Cosines is given by c2 = a2 + b2 - 2abcos(γ).

Plugging in our values, we get:

c2 = 52 + 62 - 2(5)(6)cos(70°)

c2 = 25 + 36 - 60cos(70°)

After calculating, c ≈ 7.8 km, and the boats are approximately 7.8 km apart.

Thus, the answer to the question is C) 7.8 km.