# An observer at the top of a lighthouse 55m Hugh sights a ship at a sea angle of depression of 18°. If the foot of the lighthouse is at sea level how far is the ship from the foot of the light house

## Well, well, well! Looks like someone is trying to measure some distances. Let's get to it, shall we?

So, we have an observer at the top of a lighthouse 55m high and they're peeping down at a ship. Aye, a perfect spot for a little sightseeing!

Now, the observer tells us that the angle of depression to the ship is 18°. That means the observer is looking down at an 18° angle, like they just spotted a squid doing the limbo dance. Impressive!

Since the foot of the lighthouse is at sea level, we can draw a right-angled triangle with the lighthouse, the ship, and the horizontal distance between them.

Now, if we focus on the angle of depression, we can use some trigonometric wizardry to find the distance. The tangent of the angle of depression is equal to the opposite (55m) divided by the adjacent (let's call it x, our mystery distance).

So, using a little math magic, we get:

tan(18°) = 55m / x

But wait, we can't let the math overshadow the fun! Let me just punch a few numbers into my imaginary calculator here...

Calculating... calculating... and voila! x turns out to be 55m divided by the tangent of 18°. So, if I crunched those numbers right, the ship is approximately 179.5 meters away from the foot of the lighthouse.

There you go! The ship is about 179.5 meters away from the foot of the lighthouse. Now, go grab your telescope and enjoy the view!

## To find the distance from the foot of the lighthouse to the ship, we can use trigonometry.

Let's call the distance from the foot of the lighthouse to the ship as "x".

In a right-angled triangle formed by the observer, the ship, and the foot of the lighthouse, the angle of depression is 18°.

Using the tangent function, we can set up the equation:

tan(18°) = (opposite side) / (adjacent side)

Now we can plug in the values:

tan(18°) = x / 55

To solve for x, we can rearrange the equation:

x = 55 * tan(18°)

Using a calculator, we can evaluate the expression:

x ≈ 55 * 0.3249

x ≈ 17.87

Therefore, the ship is approximately 17.87 meters away from the foot of the lighthouse.

## To find the distance between the foot of the lighthouse and the ship, we can use trigonometry and the given information.

Let's denote the distance between the foot of the lighthouse and the ship as "x".

We are given:

Height of the lighthouse (h) = 55 m

Angle of depression (θ) = 18°

Since we have a right triangle formed by the observer, the ship, and the foot of the lighthouse, we can use the tangent function to relate the angle of depression and the distance "x".

The tangent of an angle is defined as the opposite side divided by the adjacent side in a right triangle. In this case, the opposite side is the height of the lighthouse (h) and the adjacent side is the distance "x".

So, we have:

tan(θ) = opposite/adjacent

tan(θ) = h/x

Substituting the given values:

tan(18°) = 55/x

To solve for x, we can rearrange the equation:

x = 55 / tan(18°)

Using a scientific calculator or trigonometric tables, we can evaluate the tangent of 18°:

tan(18°) ≈ 0.3249

Now we can calculate the value of x:

x = 55 / 0.3249

x ≈ 169.169

Therefore, the ship is approximately 169.169 meters away from the foot of the lighthouse.

## tan18° = 55/x

take over