Question

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 16°18'. When the boat stops, the angle of depression is 48°51'. The lighthouse is 200 feet tall. How far did the boat travel from when it was first noticed until it stopped? Round your answer to the hundredths place.

Answer 1

check the picture below.

bearing in mind that there are 60 minutes in 1 degree, then 16°18' will be 16.3° and likewise 48°51' will be 48.85° if we convert those to degrees only.

$\bf y=\cfrac{200}{tan(48.85^o)}~\hspace{10em}x=\cfrac{200}{tan(16.3^o)}-y \\\\\\ x=\cfrac{200}{tan(16.3^o)}-\cfrac{200}{tan(48.85^o)}\implies x\approx 683.95 - 174.78\implies x\approx 509.17$

Answer 2

The distance traveled by the boat is 509.17 feet.

  The angle of depression from the lighthouse is the angle of elevation from the boat.  

Initially, the angle of elevation is 16°18' from the boat. With the distance from the lighthouse on the adjacent side let's say x initially, and the height of the lighthouse on the opposite side which is 200 feet, makes a right triangle. The tangent is required to calculate the distance x.

       tan (16°18) = tan(16.3°) = 200/x

       x = 200/tan(16.3°) = 683.95 feet

finally the angle of elevation is from the boat is 48°51' = 48.85°

and the final distance from lighthouse be y.

so,      y = 200/tan(48.85°) = 174.78 feet.

Hence, the distance travelled by the boat is x - y =  509.17 feet.

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