a pilot entering a bay on course 60.3 degrees at the speed of 12 knots sees a light bearing 37.3 degrees true, and 20 mins later he sees it bearing 20 degrees true. if he keeps the same course and speed,(a)when will he be nearest to the lighthouse and how near (b) when will he be 11 miles from the lighthouse?
To solve this problem, we can use trigonometry and distance-speed-time formulas. Here's how:
(a) To find when the pilot will be nearest to the lighthouse and how near, we need to find the point where the distance between the pilot and the lighthouse is minimized.
1. Draw a diagram: Draw a diagram representing the bay, the pilot's course, and the lighthouse.
2. Determine the change in the bearing: The pilot saw the light bearing 37.3 degrees true initially, and 20 mins later, it bore 20 degrees true. The change in bearing is 17.3 degrees (37.3 - 20).
3. Use trigonometry to find the distance covered during the 20 mins: We know the pilot's speed is 12 knots. The distance covered is given by the formula: distance = speed × time. So, the distance covered in 20 mins is (12 knots) × (20/60) hours = 4 nautical miles.
4. Determine the closest point to the lighthouse: Since the pilot maintains the same course and speed, the closest point to the lighthouse will be the point where the distance covered during the 20 mins is perpendicular to the lighthouse. This forms a right-angled triangle with the lighthouse as the right angle.
5. Use trigonometry to find the perpendicular distance from the lighthouse: In the right triangle formed, the side opposite to the angle of 17.3 degrees would be the perpendicular distance from the lighthouse. To find this distance, we can use the formula: perpendicular distance = distance covered × sin(change in bearing).
6. Calculate the perpendicular distance: Using the values we have, the perpendicular distance = 4 nautical miles × sin(17.3 degrees) ≈ 1 nautical mile.
Thus, the pilot will be nearest to the lighthouse after 20 mins, and the distance will be approximately 1 nautical mile.
(b) To find when the pilot will be 11 miles from the lighthouse, we need to calculate the time it takes to cover that distance.
1. Determine the speed of the pilot: The pilot's speed is given as 12 knots.
2. Use the distance-speed-time formula: To find the time to cover a specific distance, use the formula: time = distance / speed.
3. Calculate the time to cover 11 miles: Plug in the values to find the time: time = 11 miles / 12 knots ≈ 0.92 hours.
Therefore, the pilot will be approximately 11 miles from the lighthouse after around 0.92 hours.