Question

A lighthouse sits at the edge of a cliff, as shown. A ship at sea level is 610 meters from the base of the cliff. The angle of elevation from sea level to the base of

the lighthouse is 55.1°. The angle of elevation from sea level to the top of the lighthouse is 56.5°. Find the height of the lighthouse from the top of the cliff.
Do not round any intermediate computations. Round your answer to the nearest tenth.
Note that the figure below is not drawn to scale.

Answer 1

47.2 meters

Explanation:

For a right triangle, the following relation holds

`\tan \theta = (Opposite\;side)/(Adjacent\;side)`

where opposite side is the leg opposite angle θ and adjacent side is the other leg adjacent to angle θ

We are given
θ₁ = 55.1° angle of elevation to base of lighthouse

θ₂ = 56.5° angle of elevation to top of lighthouse

Let x be the height of the base from sea level

Let y be the height of the top from sea level

Therefore x - y = height of the lighthouse from the top of the cliff

Given these we can come up with two equations

`\tan(55.1^\circ) = (x)/(610) \cdots[1]\\\\\tan(56.5^\circ) = (y)/(610) \cdots[2]\\`

From equation 1:

`\tan(55.1^\circ) = (x)/(610) \\\\x = 610 * \tan(55.1^\circ)\\\\`

From equation 2

`\tan(56.5^\circ) = (y)/(610) \\\\y = 610 * \tan(56.5^\circ)`

The height of the lighthouse is therefore x - y
x - y
`x - y = 610 * \tan56.5^\circ) - 610 * \tan55.1^\circ \\\\= 610(\tan56.5^\circ - \tan55.1^\circ)\\\\= 47.1949\;7 meters`

Therefore, rounded to the nearest tenth, the height of the lighthouse from the top of the cliff is 47.2 meters