The line of sight from a small boat to the light at the top of a 35-foot lighthouse built on a cliff 15 feet above the water makes a 48° angle with the water. To the nearest foot how far is the boat from the cliff?

Question

Florisalreadytaken · Answer

|dw:1618063637953:dw|
This would be a representative image -- we first have to find what the adjacent is, and from there we can continue with the distance (the hypotenuse of the small triangle) by using the pythagoras theorem.

Florisalreadytaken · Answer

so we have

$  tan \; 48 ^o = \frac {50}{a}  $  $$  a = \frac{50}{1.111} $$ $$  a \approx  45 $$

Florisalreadytaken · Answer

that would be the distance from the boat, to THE BOTTOM of the cliff -- from here we continue with  $$ h =\sqrt{ \color{lightskyblue}{a}^2 + \color{tomato}{b}^2}  $$
$$   h =\sqrt{ \color{lightskyblue}{45}^2 + \color{tomato}{15}^2} $$
$$   h =\sqrt{ \color{lightseagreen}{2250} } $$
$$  h(\text{distance})=\color{lightseagreen}{47.4} $$
and that would be the distance from the boat to THE TOP of the cliff : D

AZ · Answer

You only need to solve for a

They just wanted to know the distance from the boat to the cliff
They didn't specify the top of the cliff or the base of the lighthouse so it's assumed that the distance they're looking for is from the boat to the base of the cliff

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