/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 44 A 200 -foot cliff drops vertical... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 44

A 200 -foot cliff drops vertically into the ocean. If the angle of elevation from 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
To calculate the exact horizontal distance, use a calculator to divide 200 by the tangent of 22.3 degrees. The result, rounded to the nearest foot, is the desired distance off shore which the ship is.

Step by step solution

01

Identify known values

From the problem, we know that the height of the cliff is 200 feet (opposite side in our right triangle) and the angle of elevation from the ship to the top of the cliff is \(22.3^{\circ}\). Thus, the known values are \(Opposite = 200\) feet and \(\theta = 22.3^{\circ}\). The distance off shore which the ship is (which is the adjacent side of the triangle) is what we're looking to find.
02

Set up the equation using tangent

We need to use the tangent function, which is the ratio of the opposite side to the adjacent side. Setting up the equation, we have: \(\tan(\theta) = \frac{Opposite}{Adjacent}\). Substituting the known values in, we get: \(\tan(22.3^{\circ}) = \frac{200}{Adjacent}\).
03

Solve for the Adjacent side

Rearranging the equation to solve for the 'Adjacent' side, we get: \(Adjacent = \frac{200}{\tan(22.3^{\circ})}\). Calculating this will provide the horizontal distance from the ship to the base of the cliff - the required distance off shore which the ship is.

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