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Chapter 10: Problem 42

A 200 -foot cliff drops vertically into the ocean. If the angle of elevation of a ship to the top of the cliff is \(22.3^{\circ}\), how far off shore, to the nearest foot, is the ship?

Short Answer

Expert verified
The ship is approximately 498 feet off the shore.

Step by step solution

01

Identify and label the given information

Identify a right triangle where the 200-foot cliff is the opposite side, the distance between the ship and the base of the cliff is the adjacent side, and the line of sight from the ship to the top of the cliff is the hypotenuse. The angle of elevation \(22.3^{\circ}\) is the angle between the adjacent side and the hypotenuse in this triangle.
02

Apply the tangent function

Because we know the angle and the length of the opposite side and we want to find the length of the adjacent side, we can use the tangent function which is defined as the ratio of the length of the opposite side to the length of the adjacent side. So, we write the equation: \(tan(22.3^{\circ}) = \frac{200}{x}\) where \(x\) is the distance off shore, to the nearest foot, is the ship.
03

Solve for the unknown

Rearrange the equation to isolate \(x\) on one side, and then solve the equation for \(x\), rounding the result to the nearest foot. This gives: \( x = \frac{200}{tan(22.3^{\circ})} \)

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