Question

Matt is standing on top of a cliff 305 feet above a lake the measurement of the angle of depression to a boat on the lake is 42 degree .how far is the boat from the base of the cliff

Answer 1

The boat is approximately 338.74 feet from the base of the cliff. If the problem suggests rounding to the nearest whole number, you should use 339 ft.

Explanation:

Let's draw the right angle triangle associated with the problem (see attached image). Notice that the 305 ft height corresponds to the "opposite side" to the given acute angle of
`42^o`, and that the distance we need to find is the "adjacent" side to the given acute angle. Therefore, the appropriate trigonometric function to use is the "tangent", given that from it we can solve for the unknown adjacent side "x":

`tan(42^o)=(opposite)/(adjacent) \\tan(42^o)=(305)/(x)\\x=(305)/(tan(42^o))\\ x=338.74\,\, ft`