Problem 44 - Algebra and Trigonometry

Question

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?

Step-by-Step Solution

Answer

The ship is approximately 498 feet from the shore.

Step 1: Identify the triangle and relevant sides

Identify the right triangle created by the cliff, the ocean, and the line of sight from the ship to the top of the cliff. Here, the cliff represents the 'opposite' side of the triangle, the ocean (distance from the ship to the cliff) represents the 'adjacent' side of the triangle, and the line of sight is the 'hypotenuse'.

Step 2: Apply Tangent function

The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. In this case, \(\tan(22.3^{\circ})\) is equal to the height of the cliff divided by the unknown distance of the ship from the base of the cliff, or \(\tan(22.3^{\circ}) = \frac{200}{x}\).

Step 3: Solve for unknown side

Rearrange the equation to solve for the unknown distance x. Multiply both sides of the equation by x, then divide by \(\tan(22.3^{\circ})\) to isolate x, the distance of the ship from the cliff. The equation is \(x = \frac{200}{\tan(22.3^{\circ})}\).

Step 4: Calculate the value

Now that we have the equation, we can plug the value into a calculator. We get \(\frac{200}{\tan(22.3^{\circ})} \approx 498\).

Step 5: Round to the nearest foot

The question asks for the distance to the nearest foot, so round the answer to the nearest whole number. This gives us a final answer of approximately 498 feet.