Problem 48 - Algebra and Trigonometry

Question

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 feet away from the base of the lighthouse.

Step 1: Understanding the Problem

We have a lighthouse that is 200 feet tall. From the top of this lighthouse, the angle of depression to a ship is given as 23 degrees. We need to find the horizontal distance from the base of the lighthouse to the ship.

Step 2: Setting Up the Right Triangle

Visualize a right triangle where the height of the lighthouse is the vertical side (opposite), the horizontal distance from the base to the ship is the adjacent side, and the line of sight to the ship is the hypotenuse. The angle between the horizontal ground line and line of sight (hypotenuse) is the angle of depression, which can also be used as the angle of elevation from the base.

Step 3: Using the Tangent Function

Recall that the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. This can be expressed as: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]For this problem, \(\theta = 23^{\circ}\), the opposite side is 200 ft, and we are solving for the adjacent side, which represents the distance to the ship.

Step 4: Calculating the Distance

We can rearrange the tangent formula to solve for the adjacent side: \[ \text{adjacent} = \frac{\text{opposite}}{\tan(\theta)} \]Plugging in the values, we get:\[ \text{adjacent} = \frac{200}{\tan(23^{\circ})} \]Using a calculator, compute \( \tan(23^{\circ}) \approx 0.4245 \), thus the distance is:\[ \text{adjacent} \approx \frac{200}{0.4245} \approx 471.000 \text{ ft} \]

Step 5: Conclusion

The horizontal distance from the base of the lighthouse to the ship is approximately 471 feet.

Key Concepts

Tangent Function

The tangent function is one of the primary trigonometric functions used in mathematics, especially in problems involving right triangles. In simpler terms, it relates the angle in a right triangle to the ratio of two of its sides. The formula to remember is:

• If you have a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
• This can be expressed as: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

In our lighthouse problem, the angle of depression is used with this function. Since the angle of depression from the top corresponds to the angle of elevation from the bottom, we can apply the tangent function directly.When solving for the horizontal distance to the ship (adjacent side), we rearranged the formula because we knew the opposite side (the lighthouse's height of 200 ft) and the angle (\(23^{\circ}\)). This function helps us bridge the gap between the known and unknown measurements in the triangle.

Right Triangle

A right triangle is a triangle in which one of the angles is exactly 90 degrees. Because of this defining feature, right triangles are widely used in various fields, including architecture, astronomy, and physics. Let's see what sets it apart:

• It has three sides: opposite, adjacent, and hypotenuse.
• The side opposite the right angle is the longest side known as the hypotenuse.
• The other two sides are simply labeled as the opposite and adjacent sides, depending on which angle you are focusing on.

In the case of the original problem, the height of the lighthouse and the ground to the ship creates a right triangle. The lighthouse serves as the opposite side to the angle of interest (angle of depression or elevation). The horizontal distance forms the adjacent side. Right triangles, like the one formed here, make it easy to calculate distances using trigonometric functions such as the tangent.

Trigonometric Ratios

Trigonometric ratios are essential tools in geometry and trigonometry, providing a bridge between angles and side lengths in right triangles. The core ratios are sine, cosine, and tangent, each of which relates different sets of triangle sides:

• **Sine** relates the opposite side to the hypotenuse.
• **Cosine** relates the adjacent side to the hypotenuse.
• **Tangent** relates the opposite side to the adjacent side, as we saw in the lighthouse problem.

These ratios are fundamental in solving real-world problems, where direct measurement of all sides might not be possible. Understanding these ratios helps in setting up equations to find unknown lengths or angles in triangles, just like how we calculated the distance to the ship. By using the tangent function, which is one of the trigonometric ratios, the problem involving an angle and a side length can be translated into a solvable equation.