a) A lighthouse point L is 10 km away from a yacht and 8 km away from a sailboat at point B. From the yacht, the lighthouse and sailboat are separated by an angle of 48 degrees. How far apart are the sailboat and yacht?

Answer 1

To find the distance between the sailboat and yacht, we can use the law of cosines. Let's call the distance between the sailboat and yacht "c".

From the given information, we know that:
- The distance between the lighthouse and yacht, which is also the distance between the lighthouse and sailboat, is 10 km.
- The distance between the lighthouse and sailboat, which is also the distance between the lighthouse and yacht, is 8 km.
- The angle between the lighthouse-sailboat line and the lighthouse-yacht line is 48 degrees.

Now, let's label the sides and angle of the triangle between the sailboat, yacht, and lighthouse:
- Side c is the distance between the sailboat and yacht (which is what we are looking for).
- Side a is the distance between the lighthouse and sailboat (8 km).
- Side b is the distance between the lighthouse and yacht (10 km).
- Angle C is the angle between side a and side b (48 degrees).

Using the law of cosines, we have:
c^2 = a^2 + b^2 - 2ab * cos(C)

Substituting the given values:
c^2 = 8^2 + 10^2 - 2 * 8 * 10 * cos(48)

Calculating the expression on the right-hand side:
c^2 = 64 + 100 - 160 * cos(48)

Now, let's calculate the value of cos(48) using a calculator:
cos(48) ≈ 0.6691306064 (rounded to 10 decimal places)

Substituting the value of cos(48) into the equation:
c^2 ≈ 64 + 100 - 160 * 0.6691306064

Simplifying the equation:
c^2 ≈ 64 + 100 - 107.062497024

c^2 ≈ 56.937502976

Taking the square root of both sides to solve for c:
c ≈ sqrt(56.937502976)

c ≈ 7.55

Therefore, the sailboat and yacht are approximately 7.55 km apart.