Question

a man on a cliff spots a ship at an angle of depression of 43°. the distance from the bottom of the cliff to the ship is 127 ft. how tall is the cliff? round to two decimal places as needed.

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the angle of depression is \(43^\circ\), the adjacent side to this angle (in the right triangle formed by the cliff, the horizontal, and the line of sight) is the distance from the bottom of the cliff to the ship (\(127\) ft), and the opposite side is the height of the cliff (\(h\)). The angle of depression is equal to the angle of elevation from the ship to the man, so we can use the tangent function: \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\).

Step2: Set up the equation

Let \(\theta = 43^\circ\), adjacent \(= 127\) ft, opposite \(= h\). So \(\tan(43^\circ)=\frac{h}{127}\).

Step3: Solve for \(h\)

Multiply both sides by \(127\): \(h = 127\times\tan(43^\circ)\). Calculate \(\tan(43^\circ)\approx0.9325\), then \(h = 127\times0.9325\approx118.43\) (rounded to two decimal places).

Answer:

The height of the cliff is approximately \(118.43\) feet.