The captain of a ship at sea sights a lighthouse which is 120 feet tall.

The captain measures the the angle of elevation to the top of the lighthouse to be 19 ^\circ.

How far is the ship from the base of the lighthouse?

H = Hight of lighthouse

L = Distance between a ship and a lighthouse

tan( theta ) = H / L

If 19^ mean 19° then:

tan( theta ) = H / L

tan( 19° ) = 120 / L

L * tan( 19° ) = 120 Divide both sides with tan( 19° )

L = 120 / tan ( 19° )

L = 120 / 0.34433

L = 348.5 ft

If 19^ mean 19´ then:

tan( theta ) = H / L

tan( 19´ ) = 120 / L

L * tan( 19´ ) = 120 Divide both sides with tan( 19´ )

L = 120 / tan ( 19´ )

L = 120 / 0.00553

L = 21,699.82 ft

If 19^ mean 19" then:

tan( theta ) = H / L

tan( 19" ) = 120 / L

L * tan( 19" ) = 120 Divide both sides with tan( 19" )

L = 120 / tan ( 19" )

L = 120 / 0.00009

L = 1,333,333.33 ft

The captain of a ship at sea sights a lighthouse which is 160 feet tall. The captain measures the the angle of elevation to the top of the lighthouse to be 17 °. How far is the ship from the base of the lighthouse?

Well, first of all, it's nice to see a captain with good vision! Now, let's figure out how far the ship is from the base of the lighthouse. We can use some trigonometry here!

We know the height of the lighthouse is 120 feet, and the angle of elevation to the top of the lighthouse is 19 degrees. So, we can use the tangent function to find the distance.

Tangent(angle) = Opposite/Adjacent

In this case, the angle is 19 degrees, the opposite side is 120 feet (height of the lighthouse), and we're looking for the adjacent side (distance from the ship to the base of the lighthouse).

So, let's plug those values in:

Tangent(19) = 120/Adjacent

Now, let's solve for Adjacent:

Adjacent = 120 / Tangent(19)

Now, if you want the answer in feet, just plug this into your calculator and you'll get your distance. If you're looking for a more practical measurement, then turn the ship around and sail directly to the lighthouse!

To find the distance between the ship and the base of the lighthouse, we can use trigonometry. Specifically, we can use the tangent function.

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

Using the given information, we have the height of the lighthouse (120 feet) and the angle of elevation to the top of the lighthouse (19 degrees).

The tangent function relates the angle of elevation (19 degrees) to the opposite side (120 feet) and the adjacent side ('x').

The tangent of an angle is equal to the opposite side divided by the adjacent side:

tan(angle) = opposite / adjacent

In this case:

tan(19 degrees) = 120 / x

We can now solve this equation for 'x' by rearranging it:

x = 120 / tan(19 degrees)

Calculating this value, we get:

x ≈ 363.28 feet

Therefore, the ship is approximately 363.28 feet away from the base of the lighthouse.