Answer:o calculate the distance between the ship and the lighthouse, we can use the Pythagorean Theorem.
a) The navigator measures ZTSL, which is the angle between the ship (T) and the lighthouse (L). Since the ship sails 19.5 km north to point S, we can consider the distance from T to S as the perpendicular side of a right triangle. The hypotenuse of the triangle is the distance between the ship and the lighthouse, which we want to find.
b) We know that ZTSL is 33°, and the length of TS is 19.5 km. Using the Pythagorean Theorem, we can calculate the distance between the ship and the lighthouse (LT).
Let's calculate it step by step:
1. Identify the sides of the right triangle:
- The side opposite the right angle is LT.
- The side adjacent to the angle ZTSL is TS.
2. Apply the Pythagorean Theorem:
LT^2 = TS^2 + LS^2
3. Substitute the known values:
LT^2 = 19.5^2 + LS^2
4. Solve for LT:
LT = sqrt(19.5^2 + LS^2)
Since we know that LS is the distance from T to S, we can calculate it by using the trigonometric function sine:
LS = TS * sin(ZTSL)
5. Substitute the value of LS in the equation:
LT = sqrt(19.5^2 + (19.5 * sin(33))^2)
6. Simplify and calculate the result:
LT ≈ 12.7 km
Therefore, the distance between the ship and the lighthouse is approximately 12.7 km.
Explanation: