A boat is 23 mi due west of lighthouse A. Lighthouse B is 14 mi due north of lighthousenA. Find the bearing of lighthouse B from the boat and the distance from lighthousenB tho the boat.

To find the bearing of lighthouse B from the boat, we need to determine the angle between the line connecting the boat and lighthouse B, and the line pointing due north.

First, let's draw a diagram to visualize the situation.

Lighthouse B
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| boat
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| Lighthouse A
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From the given information, we can see that the boat is 23 miles due west of lighthouse A, and lighthouse B is 14 miles due north of lighthouse A.

Using basic trigonometry, we can calculate the answer.

1. Find the distance between the boat and lighthouse B:
- This can be calculated using the pythagorean theorem. The horizontal distance from the boat to lighthouse B is the same as the distance between the boat and lighthouse A, which is 23 miles. The vertical distance from lighthouse B to lighthouse A is 14 miles.
- Using the pythagorean theorem: Distance^2 = horizontal distance^2 + vertical distance^2
- Distance^2 = 23^2 + 14^2
- Distance^2 = 529 + 196
- Distance^2 = 725
- Distance ≈ 26.91 miles (rounded to two decimal places)

2. Find the bearing of lighthouse B from the boat:
- To find the bearing, we need to calculate the angle between the line connecting the boat and lighthouse B, and the line pointing due north.
- We can use inverse trigonometric functions to calculate this angle.
- The angle can be found using the tangent of the angle: tan(angle) = vertical distance / horizontal distance
- tan(angle) = 14 / 23
- angle ≈ 29.74 degrees (rounded to two decimal places)

Therefore, the bearing of lighthouse B from the boat is approximately 29.74 degrees, and the distance from lighthouse B to the boat is approximately 26.91 miles.