The line of sight from a small boat to the light at the top of a 45-foot lighthouse built on a cliff 25 feet above the water makes a 20° angle with the water. to the nearest foot, how far is the boat from the cliff?

The boat is 192 feet from the cliff.

Explanation:

Hi there!

Please see the attached figure for a graphical description of the problem.

Notice that the line of sight, the distance to the cliff and the height to the top of the lighthouse form a right triangle. Hence, we can apply trigonometric rules to find the distance from the boat to the cliff:

cos 20° = adjacent side / hypotenuse

sin 20° = opposite side / hypotenuse

The length of the opposite side is the height of the cliff plus the height of the lighthouse:

opposite side = 45 feet + 25 feet = 70 feet.

Using the equation of sin 20°, we can obtain the hypotneuse:

sin 20° = opposite side / hypotenuse

hypotenuse · sin 20° = opposite side

hypotenuse = opposite side / sin 20°

hypotenuse = 70 feet / sin 20°

hypotenuse = 205 feet

Now, using the equation of cos 20°, we can calculate the distance to the cliff (the length of the adjacent side):

cos 20° = adjacent side / hypotenuse

hypotenuse · cos 20° = adjacent side

205 feet · cos 20° = adjacent side

adjacent side = 192 feet  (without rounding intermediate results)

The boat is 192 feet from the cliff.

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