Question

An observer on top of a 50-foot tall lighthouse sees a boat at a 7° angle of depression. To the nearest foot, how far is the boat from the base of the lighthouse?

Answer 1

Answer:

407.22 foot is the boat from the base of the lighthouse

Step-by-step explanation:

Given the statement: An observer on top of a 50-foot tall lighthouse sees a boat at a 7° angle of depression.

Let x foot be the distance of the object(boat) from the base of the lighthouse

Angle of depression = $7^{\circ}$

$\angle CAB = \angle DCA = 7^{\circ}$       [Alternate angle]

In triangle CAB:

To find AB = x foot.

Using tangent ratio:

$\tan (\theta) = \frac{Opposite side}{Adjacent Base}$

$\tan (\angle CAB) = \frac{BC}{AB}$

Here, BC = 50 foot and $\angle CAB =7^{\circ}$

then;

$\tan (7^{\circ}) = \frac{50}{x}$

or

$x = \frac{50}{\tan 7^{\circ}}$

$x = \frac{50}{0.1227845609}$

Simplify:

AB = x = 407.217321 foot

Therefore, the boat from the base of the light house is, 407.22'