The captain of a ship at point A and sailing toward point B observes a lighthouse at L and finds angle LAC to be 36*30'. After sailing 5 km to B, he observes angle LBC to be 73*. How many kilometers is position B from the lighthouse?
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To solve this problem, we can use the Law of Sines. The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of the opposite angle is constant.
Let's break down the problem and label the given information:
- The captain is at point A.
- The captain observes the lighthouse at point L.
- The captain sails to point B, which is 5 km away from point A.
- The captain measures angle LAC to be 36 degrees 30 minutes (36°30').
- The captain measures angle LBC to be 73 degrees (73°).
We want to find the distance between point B and the lighthouse at point L.
First, let's find the length of side AC. Since we know the measure of angle LAC and side AC is opposite to this angle, we will use the Law of Sines:
sin(LAC) / AC = sin(BCA) / BC
In this equation:
- sin(LAC) is the sine of angle LAC (which we can convert to decimal degrees),
- AC is the unknown side length,
- sin(BCA) is the sine of angle BCA (which is opposite to side AC), and
- BC is the length of side BC (which is 5 km).
Let's plug in the values we have:
sin(36°30') / AC = sin(73°) / 5
To solve for AC, rearrange the equation:
AC = (sin(36°30') * 5) / sin(73°)
Now, you can use a scientific calculator to find the sine of 36°30' and the sine of 73°. Multiply them by 5, and divide the result of the numerator by the result of the denominator to get the value of AC.
Once you have the value of AC (the length from the lighthouse to point A), you can determine the position of the lighthouse from point B by subtracting AC from the total distance sailed by the captain (5 km).
Final Answer:
The position B is (5 km - AC) km away from the lighthouse.