Problem 50 - Algebra and Trigonometry

Question

Distance at Sea From the top of a 200 -ft lighthouse, the angle of depression to a ship in the ocean is \(23^{\circ} .\) How far is the ship from the base of the lighthouse?

Step-by-Step Solution

Answer

The ship is approximately 471.25 feet from the lighthouse base.

Step 1: Understanding the Problem

In this problem, we have a lighthouse and a ship. The lighthouse is 200 feet tall, and the angle of depression from the top of the lighthouse to the ship is \(23^{\circ}\). We need to find the horizontal distance from the ship to the base of the lighthouse.

Step 2: Setting Up the Triangle

We can use trigonometry to solve this problem by imagining a right triangle formed by the lighthouse height, the line of sight to the ship, and the horizontal distance from the lighthouse base to the ship. The angle of depression \(23^{\circ}\) is equivalent to the angle of elevation from the ground up to the ship's line of sight.

Step 3: Applying Trigonometric Ratios

In this right triangle, the 200-ft lighthouse is the opposite side to the angle, and the distance from the base to the ship is the adjacent side. We can use the tangent function, which relates the opposite and adjacent sides: \( \tan(23^{\circ}) = \frac{200}{d} \), where \(d\) is the distance from the base of the lighthouse to the ship.

Step 4: Solving for the Distance

Rearrange the equation to solve for \(d\): \(d = \frac{200}{\tan(23^{\circ})}\). Calculate \( \tan(23^{\circ}) \) using a calculator and then find \(d\).

Step 5: Calculating the Distance

First, find \( \tan(23^{\circ}) \approx 0.4245\). Then substitute back into the equation to find \(d\): \(d = \frac{200}{0.4245} \approx 471.25 \). Thus, the ship is approximately 471.25 feet from the base of the lighthouse.

Key Concepts

Angle of Depression

The angle of depression is a crucial concept in trigonometry and navigation. It is the angle formed between the horizontal line from the observer's eye and the line of sight down to an object. In our problem, the lighthouse keeper observes a ship; the angle of depression is given as \(23^{\circ}\).
When solving problems like these, it's useful to remember that the angle of depression from the observer's point is equal to the angle of elevation from the target's viewpoint, due to the alternate interior angles formed by the parallel lines. This means, if you are calculating from the ground, you could use the same angle but call it the angle of elevation instead.
Understanding these angles helps you find distances such as how far an object like the ship is away from a point on the ground, using trigonometric functions.

Tangent Function

The tangent function is a key trigonometric function used to relate different sides of a right triangle. Specifically, in a right triangle, the tangent (\(\tan\)) of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
In our exercise, the opposite side is the 200-ft height of the lighthouse, while the adjacent side is the unknown distance from the base of the lighthouse to the ship. Thus, we use

• \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)

For this problem, substituting the given values gives us:
\(\tan(23^{\circ}) = \frac{200}{d}\).
Solving such equations allows us to find distances or heights, making tangent a valuable tool in navigation and other fields where measurement and distance are important.

Right Triangle

A right triangle is a triangle in which one of the angles is exactly \(90^{\circ}\). Such triangles are pivotal in trigonometry, as they form the basis for many trigonometric functions and equations. Understanding the properties of right triangles is essential for solving distance and angle problems, such as in our lighthouse and ship scenario.
In this situation, our right triangle is formed by:

• The height of the lighthouse as the vertical side.
• The distance from the base of the lighthouse to the ship as the horizontal side.
• The hypotenuse is the line of sight from the top of the lighthouse to the ship.

These components show how trigonometric ratios can be applied to find the unknown distance, using the given height and angle information. Always remember, the right triangle supports the tangent function by allowing us to explore angles and sides relationships effectively.