lighthouse b is 9 miles west of lighthouse A. a boat leaves A and sails 5 miles. At this time, it is sighted from B. If the bearing of the boat from B is N64 E, how far from B is the boat?

To find the distance from the boat to lighthouse B, we can use trigonometry. The bearing N64 E can be converted into an angle relative to the north direction.

The bearing N64 E means the boat is traveling along a line that is 64 degrees east of north. In trigonometry, we measure angles clockwise from the positive y-axis, so we need to subtract 64 degrees from 90 degrees to get the angle relative to the north direction.

Angle relative to north = 90 degrees - 64 degrees = 26 degrees

Now, we can use trigonometry to find the distance from the boat to lighthouse B. We have a right-angled triangle with the boat's distance from B as the hypotenuse, the distance traveled by the boat as the adjacent side, and the angle relative to north as the reference angle.

Using the cosine function:

cos(angle relative to north) = adjacent / hypotenuse

cos(26 degrees) = 5 miles / hypotenuse

Solving for the hypotenuse (distance from the boat to lighthouse B):

hypotenuse = 5 miles / cos(26 degrees)

Using a calculator, we can find:

hypotenuse ≈ 5 miles / 0.8944 ≈ 5.588 miles

Therefore, the boat is approximately 5.588 miles from lighthouse B.

To find the distance between the boat and lighthouse B, we can use trigonometry and the given information. Let's break down the problem step by step:

1. Start by drawing a diagram. Draw lighthouse A and lighthouse B, with lighthouse B located 9 miles west of lighthouse A.

2. From lighthouse A, draw a line representing the path the boat takes, sailing 5 miles.

3. From lighthouse B, draw a line representing the line of sight to the boat. The bearing given, N64 E, means that the angle between the line of sight and the north direction is 64 degrees, and the angle between the line of sight and the east direction is 90 - 64 = 26 degrees.

4. Now we have a triangle formed by lighthouse B, the boat, and the line of sight from B. Since we know the angle between the line of sight and the east direction (26 degrees), and we know the side adjacent to this angle (the distance sailed by the boat, 5 miles), we can use trigonometry.

5. We can apply the cosine function to find the distance between the boat and lighthouse B. The cosine of the angle (26 degrees) is equal to the adjacent side (distance between the boat and lighthouse B) divided by the hypotenuse (unknown distance we want to find).

6. Rearranging the equation, the distance between the boat and lighthouse B is equal to the distance sailed by the boat (5 miles) divided by the cosine of the angle (26 degrees).

7. Calculate the cosine of 26 degrees. You can use a scientific calculator or an online calculator to find the cosine of the angle.

8. Divide the distance sailed by the boat (5 miles) by the cosine of 26 degrees to find the distance between the boat and lighthouse B.

Once the calculation is done, you'll have the answer to the question, which is the distance from lighthouse B to the boat.

One way to do this is to set up coordinates.

Let A = (0,0)
B = (-9,0)

When A has sailed 5 miles, it lies on the circle

x^2+y^2 = 25

The line through B with slope 0.4877 is
y = .4877(x+9)

This line intersects the circle at two places: (-4.5,2.2) and (1.0,4.9)

So, you gotta be asking yourself, "Do I feel lucky?"