A ship
A ship has been spotted by two lighthouses, A and B, as shown in the figure. What is the distance from the ship to Lighthouse A to the nearest tenth? Figure - the distance between lighthouses A and B is 40 nautical miles. From A is seen in view angle 57° and from B at 64° angle.
Correct answer:
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Dr. Math
Solution:
To find the distance from the ship to Lighthouse A, we can use the Law of Sines. The given information is:
- The distance between Lighthouse A and Lighthouse B is 40 nautical miles.
- The angle at Lighthouse A is 57° .
- The angle at Lighthouse B is 64° .
Step 1: Find the angle at the ship.
The sum of angles in a triangle is 180° . Therefore, the angle at the ship ( C ) is:
Step 2: Use the Law of Sines to find the distance from the ship to Lighthouse A.
The Law of Sines states:
Here:
- a is the distance from the ship to Lighthouse B,
- b is the distance from the ship to Lighthouse A (the value we are solving for),
- c = 40 nautical miles (the distance between Lighthouse A and Lighthouse B),
- A = 57° (the angle at Lighthouse A),
- B = 64° (the angle at Lighthouse B),
- C = 59° (the angle at the ship).
Using the Law of Sines:
Solve for b :
Final Answer:
The distance from the ship to Lighthouse A is approximately:
To find the distance from the ship to Lighthouse A, we can use the Law of Sines. The given information is:
- The distance between Lighthouse A and Lighthouse B is 40 nautical miles.
- The angle at Lighthouse A is 57° .
- The angle at Lighthouse B is 64° .
Step 1: Find the angle at the ship.
The sum of angles in a triangle is 180° . Therefore, the angle at the ship ( C ) is:
C = 180° - 57° - 64° = 59°
Step 2: Use the Law of Sines to find the distance from the ship to Lighthouse A.
The Law of Sines states:
asin A = bsin B = csin C
Here:
- a is the distance from the ship to Lighthouse B,
- b is the distance from the ship to Lighthouse A (the value we are solving for),
- c = 40 nautical miles (the distance between Lighthouse A and Lighthouse B),
- A = 57° (the angle at Lighthouse A),
- B = 64° (the angle at Lighthouse B),
- C = 59° (the angle at the ship).
Using the Law of Sines:
bsin 64° = 40sin 59°
Solve for b :
b = 40 · sin 64°sin 59°
b = ≈ 41.9
Final Answer:
The distance from the ship to Lighthouse A is approximately:
41.9 nautical miles
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Do you want to convert length units?
See also our trigonometric triangle calculator.
Try conversion angle units angle degrees, minutes, seconds, radians, grads.
You need to know the following knowledge to solve this word math problem:
planimetricsbasic operations and conceptsgoniometry and trigonometryUnits of physical quantitiesGrade of the word problem
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