A person is watching a boat from the top of a lighthouse. The boat is approaching the lighthouse directly. When first noticed the angle of depression to the boa

Question

Help me please and show work please 

Answer 1

I'm going to assume the question is

"A person is watching a boat from the top of a lighthouse. The boat is approaching the lighthouse directly. When first noticed, the angle of depression to the boat is θ₁ degrees. When the boat stops, the angle of depression is θ₂ degrees. The lighthouse is h feet tall. How far did the boat travel from when it was first noticed until it stopped? Round your answer to the hundredths place." You will have to plug in the values for angles θ₁ and θ₂, and the height h.

Let's assume the lighthouse stands straight up, and that the surface ofd the sea is horizontal. Then when the boat is first sighted, we have a right triangle whose three vertices are the base of the lighthouse, the observer, and the boat. Draw this triangle with the right angle (the base of the lighthouse) at the lower left, and the observer (on top of the lighthouse) at the top. The size of the angle in the triangle where the observer is located is 90-θ₁ degrees. The tangent of this angle is "opposite over adjacent". So

tan (90-θ₁) = (old distance from boat to base)/h

Multiplying both sides of the equation by h,

h·tan (90-θ₁) = (old distance from boat to base)

A few minutes later, the boat is closer to the base of the lighthouse. We have a new right triangle whose three vertices are the base of the lighthouse, the observer, and the boat. This time, the size of the angle in the triangle where the observer is located is 90-θ₂ degrees. So

tan (90-θ₂) = (new distance from boat to base)/h

Multiplying both sides of the equation by h,

h·tan (90-θ₂) = (new distance from boat to base)

The distance the boat traveled between the two sightings is therefore

(old distance from boat to base) - (new distance from boat to base)

= (h·tan (90-θ₁)) - (h·tan (90-θ₂))

which you can simplify to

h((tan (90-θ₁)) - (tan (90-θ₂))).

Now plug in the values for θ₁ degrees, θ₂ degrees, and h feet, and round the answer as required by the problem. Hope this helped.