A ship is sighted directly east of a lighthouse. Another ship, which is 20m away from the first ship, is observed at a bearing of N25degreesE from the lighthouse. If the first ship is 4.1 km away from the lighthouse, what is the distance of the second ship from the lighthouse?

angle at ship B

(sin 25)/20 = (sin A)/4.1
sin A = .0866
A = 4.97 deg
angle at ship A
A = 180 - 25 - 4.97 = 150.03 degrees
then law of cosines
d^2 = 20^2 + 4.1^2 - 2(20)(4.1)cosA

To solve this problem, we can use basic trigonometry. Let's break down the given information:

1. The first ship is directly east of the lighthouse.
2. The second ship is 20m away from the first ship.
3. The second ship is observed at a bearing of N25degreesE from the lighthouse.
4. The first ship is 4.1 km away from the lighthouse.

Since the first ship is directly east of the lighthouse, we can consider it as the reference point. Now, let's draw a diagram to visualize the situation:

Ship 1 Lighthouse Ship 2
/
N25°E
/
/
/
/
/
/
/
/
/
/
/
/
/
/

From the diagram, we can see that we have a right-angled triangle formed by the lighthouse, the first ship, and the second ship. The distance between the lighthouse and the second ship is the hypotenuse of this triangle.

Now, let's use trigonometry to find the distance between the lighthouse and the second ship. We have the following information:

- We know the length of the adjacent side (the distance between the lighthouse and the first ship) is 4.1 km.
- We know the angle between the lighthouse, the first ship, and the second ship is 25 degrees.

To find the length of the opposite side (the distance between the first and second ships), we can use the trigonometric function tangent, which is defined as:

tan(angle) = opposite / adjacent

Rearranging the formula, we have:

opposite = tan(angle) * adjacent

Plugging in the values, we have:

opposite = tan(25°) * 4.1 km

Using a scientific calculator, we find:

tan(25°) ≈ 0.46630765815

Therefore, the distance between the lighthouse and the second ship is:

opposite ≈ 0.46630765815 * 4.1 km

Calculating this multiplication, we get:

opposite ≈ 1.911 km

So, the distance of the second ship from the lighthouse is approximately 1.911 km.