Question

A boat is heading towards a lighthouse, where Meena is watching from a vertical

distance of 120 feet above the water. Meena measures an angle of depression to the
boat at point A to be 18º. At some later time, Meena takes another measurement and
finds the angle of depression to the boat (now at point B) to be 44º. Find the distance
from point A to point B. Round your answer to the nearest foot if necessary.

Answer 1

245 ft

Explanation:

The geometry of the problem can be modeled by a right triangle in which Meena's height is the side opposite the angle of depression, and the distance to the boat is the side adjacent to the angle of depression. The mnemonic SOH CAH TOA reminds us of the relation ...

Tan = Oppsite/Adjacent

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The more useful trig function for this problem is the inverse of the tangent function, the cotangent. Using that function, we have ...

cot(angle) = (distance to boat)/(Meena's height)

Then the distance to the boat is ...

distance to boat = (Meena's height)×cot(angle)

The change in distance from A to B is then ...

A -B = (120 ft)(cot(18°) -cot(44°))

= (120 ft)(3.0777 -1.0355) ≈ 245.06 ft

The distance from point A to point B is about 245 feet.