From a lighthouse (270m above the ocean), an observer notices a ship. The lighthouse is setback from the ocean about 60m. To keep the ships safe from an underwater obstacle at this spot, they need to stay at least 1.4km away from shore. To see the ship, the observer must look down through an angle of depression of 3.24 degrees. How close is the ship to hitting the obstacle?

I assume that by "at this spot" you mean 1.4km from the shore at the base of the lighthouse.

So, draw a diagram.

Let T be the top of the lighthouse
Let B be the base of the lighthouse
Let S be the point on the shore 60 form B
Let V be the location of the vessel (or ship, but S was taken)
Let O be the obstacle

We want to figure OV = BV - BO

SO = 1.4km = 1400m
tan 3.24° = TB/BV = 270/BV, so BV = 4769
SV = BV-BS = 4769-60 = 4709

The ship is 4709-60 = 4649m from shore, and thus 4649-1400m or 3249m or 3.25km from the obstacle

To determine how close the ship is to hitting the obstacle, we need to use trigonometry and the given information.

Let's break down the problem step by step:

Step 1: Draw a diagram
Draw a diagram to visualize the problem. The lighthouse is located 270m above the ocean and 60m back from the shore.

O
/|
/ |
/ | 270m
/ |
/ |
/ |
/___|\
// 60m \
//_________\
Ship x Lighthouse

Step 2: Identify the relevant angles
The angle of depression, which is the angle between the line of sight from the observer to the ship and the horizontal line, is given as 3.24 degrees.

Step 3: Set up the trigonometric relationship
We have a right-angled triangle formed by the observer, the ship, and the obstacle. We can use the tangent function to relate the angle of depression, the height of the lighthouse, and the distance from the ship to the obstacle.

Since we know the height of the lighthouse above the ocean (270m) and the distance of the ship from the lighthouse (x + 60m), we can write the equation:

tan(angle of depression) = (height of lighthouse) / (distance from the ship to the obstacle)

tan(3.24 degrees) = 270m / (x + 60m)

Step 4: Solve the equation
To find x, we can rearrange the equation and solve for it.

x + 60m = (270m) / tan(3.24 degrees)

x = (270m) / tan(3.24 degrees) - 60m

Using a calculator, we can find the value of x:

x ≈ 22646.61 meters

Step 5: Convert the answer to kilometers
To express the distance in kilometers, we divide by 1000:

x ≈ 22.64661 kilometers

Therefore, the ship is approximately 22.64661 kilometers away from hitting the obstacle.