From the top of a 50-m high bridge, two boats are seen at anchor. One boat can be seen at

an angle of depression of 38°. The other boat has a 35° angle of depression. The two
boats are separated by an angle of 110⁰. How far apart are the boats?

I will assume that 110 deg angle between boats is measured with a bearing compass in the horizontal plane and not by a tilted sextant.

How far is the first one from the bridge in water horizontal plane?
tan 38 = 50/a so a = 64 m
second one
tan 35 = 50/b so b = 71.4 m
law of cosines
cos 110 = - 0.342
c^2 = a^2 + b^2 - 2 a b cos 110
solve for c

Well, it seems like those boats are really enjoying their game of "Hide and Sea-k." Let's do some math to figure out how far apart they are!

First, we need to find out the distance from the top of the bridge to each boat. Let's call the distance to the first boat "x" and the distance to the second boat "y".

For the first boat, we can use the angle of depression of 38°. Since the angle of depression is measured from the horizontal, we can use trigonometry to find the distance. The tangent of 38° is equal to the opposite side (50m) divided by the adjacent side (x). So, we have:

tan(38°) = 50/x

Similarly, for the second boat, we can use the angle of depression of 35°. Again, using trigonometry, we get:

tan(35°) = 50/y

Now, we have a tricky part. We need to find the distances x and y, but we also know that the angle between the two boats is 110°. This means that the sum of the angles made by the triangle formed by the top of the bridge and the two boat locations is 180°. So, we can write an equation:

38° + 35° + 110° = 180°

This will help us solve for the missing angles and eventually find the distances x and y.

Once we find x and y, we can simply subtract them to get the distance between the two boats.

I hope these calculations don't make you feel too "all at sea"!

To find the distance between the two boats, we can use trigonometry and the angles of depression given.

Let's consider the boat with the 38° angle of depression first. We can label this boat as Boat A.

The angle of depression is the angle formed between the horizontal line and the line of sight from the observer (standing on the bridge) to the object (Boat A).

The height of the bridge, which is given as 50 meters, represents the vertical distance from the observer to Boat A.

We can use the tangent function to find the horizontal distance between the observer and Boat A.

Using the tangent function, we have:

tan(38°) = opposite/adjacent
tan(38°) = 50 meters/adjacent

Now, let's solve for the adjacent side:

adjacent = 50 meters / tan(38°)

Using a calculator, we find:

adjacent ≈ 50 meters / 0.7813

adjacent ≈ 63.98 meters

So, the horizontal distance between the bridge and Boat A is approximately 63.98 meters.

Next, let's consider the boat with the 35° angle of depression. We can label this boat as Boat B.

Using the same logic as before, we can use the tangent function to find the horizontal distance between the observer and Boat B.

tan(35°) = opposite/adjacent
tan(35°) = 50 meters/adjacent

Again, using a calculator, we find:

adjacent = 50 meters / tan(35°)

adjacent ≈ 50 meters / 0.7002

adjacent ≈ 71.39 meters

So, the horizontal distance between the bridge and Boat B is approximately 71.39 meters.

Now, the problem states that the two boats are separated by an angle of 110°.

To find the distance between the two boats, we can use the Law of Cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles.

Using the Law of Cosines, we have:

c^2 = a^2 + b^2 - 2ab*cos(C)

Where:
- c represents the distance between the two boats
- a and b represent the distances from the bridge to each boat
- C represents the angle between the distances a and b

Plugging in the given values, we have:

c^2 = (63.98 meters)^2 + (71.39 meters)^2 - 2 * (63.98 meters) * (71.39 meters) * cos(110°)

Now, let's solve for c:

c = sqrt((63.98 meters)^2 + (71.39 meters)^2 - 2 * (63.98 meters) * (71.39 meters) * cos(110°))

Using a calculator, we find:

c ≈ sqrt(4095.0404 + 5090.0321 - 2 * 63.98 * 71.39 * cos(110°))

c ≈ sqrt(4095.0404 + 5090.0321 - 2 * 63.98 * 71.39 * -0.34202)

c ≈ sqrt(4095.0404 + 5090.0321 + 1534.143)

c ≈ sqrt(10719.2155)

c ≈ 103.53 meters

Therefore, the two boats are approximately 103.53 meters apart.

To determine the distance between the two boats, we can use trigonometric ratios in the right-angled triangles formed by the boats and the bridge.

Let's start by reviewing the given information and labeling the diagram:
- The height of the bridge is 50 m.
- One boat has an angle of depression of 38°, which means the line of sight from the top of the bridge to the boat is 38° below the horizontal line.
- The other boat has an angle of depression of 35°.
- The two boats are separated by an angle of 110°.

Now, let's break down the problem into two separate right-angled triangles.

Triangle 1:
In this triangle, the height is the difference between the height of the bridge and the height of the boat with an angle of depression of 38°. Let's call this height "h1," and h1 = 50 m - height of the boat.

Triangle 2:
Similarly, in this triangle, the height is the difference between the height of the bridge and the height of the boat with an angle of depression of 35°. Let's call this height "h2," and h2 = 50 m - height of the boat.

Next, we can use the tangent ratio to find the distances from the bridge to each boat.

In Triangle 1: tan(38°) = h1 / x1, where x1 is the distance from the bridge to the first boat.

In Triangle 2: tan(35°) = h2 / x2, where x2 is the distance from the bridge to the second boat.

Rearranging the equations, we get:
x1 = h1 / tan(38°) and x2 = h2 / tan(35°).

To find the separation distance between the two boats, we need to calculate the horizontal distance between the two boats. We can achieve this by subtracting x2 from x1, as the boats are separated horizontally.

Separation distance = x1 - x2 = (h1 / tan(38°)) - (h2 / tan(35°)).

Finally, we substitute the given values for h1 and h2 (50 m minus the height of each boat), calculate the trigonometric ratios, and subtract to find the separation distance.