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 mak
es a 20° angle with the water. To the nearest foot, how far is the boat from the cliff?
2 answers:
Answer:
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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