Problem 20 - Contemporary Precalculus

Question

From the top of a 130 -foot-high lighthouse, the angle of depression to a boat in Lake Erie is \(2.5^{\circ} .\) How far is the boat from the lighthouse?

Step-by-Step Solution

Answer

Answer: The boat is approximately 2975.28 feet away from the lighthouse.

Step 1: Draw a diagram

Draw a right-angled triangle where one of the angles is the angle of elevation(probably \(A\)), the height of the lighthouse is the vertical leg(probably \(a\)), and the distance between the boat and the lighthouse is the horizontal leg(probably \(b\)).

Step 2: Determine the given measurements

We are given the angle of depression from the lighthouse to the boat, which is the same as the angle of elevation from the boat to the top of the lighthouse (\(A = 2.5^{\circ}\)). The lighthouse height is given, which is the vertical leg of our triangle (\(a = 130\) feet).

Step 3: Use the tangent function

We can use the tangent function to find the distance between the boat and the lighthouse, which is the horizontal leg of our triangle: \(\tan(A) = \frac{a}{b}\). Plug in the given values for \(A\) and \(a\) in the formula: \(\tan(2.5^{\circ}) = \frac{130 \text{ ft}}{b}\).

Step 4: Solve for the horizontal leg

To solve for the horizontal leg \(b\), multiply both sides of the equation by \(b\) to isolate the variable: \(b \times \tan(2.5^{\circ}) = 130 \text{ ft}\). Finally, divide both sides of the equation by \(\tan(2.5^{\circ})\) to get the value for \(b\): \(b = \frac{130 \text{ ft}}{\tan(2.5^{\circ})}\).

Step 5: Calculate the distance

Now, plug \(2.5^{\circ}\) into a calculator to find the tangent value, and then compute the distance: \(b = \frac{130 \text{ ft}}{\tan(2.5^{\circ})} \approx 2975.28 \text{ ft}\).

Step 6: State the answer

The boat is approximately 2975.28 feet away from the lighthouse.

Key Concepts

Right Angle Triangle

In geometry, a right-angle triangle is a fundamental shape that includes one angle of exactly 90 degrees. This particular angle creates a unique type of triangle that allows for various calculations using the other two angles, which sum up to 90 degrees as well. In a right-angle triangle, knowing any angle besides the right angle can provide profound insights into the properties of the triangle. For example, using trigonometric functions, you can calculate side lengths or unknown angles.

A right-angle triangle comprises:

• The 'opposite' side, which is across from the angle you are considering.
• The 'adjacent' side, which is next to the angle you are considering and is not the hypotenuse.
• The 'hypotenuse', which is the longest side of the triangle, opposite the right angle.

In problems involving right-angle triangles, such as the lighthouse example, it's common to draw a diagram first. This helps visualize and identify key parts of the triangle, such as angles of interest and sides relevant to calculation.

Angle of Depression

The angle of depression is a specific angle formed between the horizontal line from an observer and their line of sight downwards to an object. In problems relating to trigonometry, such as the lighthouse observing the boat, the angle of depression helps determine distances or heights.

It's important to note that the angle of depression is equal to the angle of elevation, which is the angle going upwards from the object back to the observer. Therefore:

• The angle of depression and the angle of elevation are the same when drawn from the observer's eye and the object's point back respectively.
• This equality is due to the horizontal line being parallel when sightlines measure angles of depression and elevation, making the angles alternate interior angles.

In our lighthouse example, the angle of depression from the top of the lighthouse to the boat is used to find the distance directly between the base of the lighthouse and the boat using trigonometric functions.

Tangent Function

Among the trigonometric functions, the tangent function stands out when dealing with right angles and involves the ratio of two sides of a right-angle triangle. Specifically, the tangent of an angle in a right-angle triangle is the ratio of the length of the opposite side to the length of the adjacent side.For example, in the lighthouse problem:

• The vertical height of the lighthouse is the 'opposite' side in relation to the angle of depression.
• The distance we want to find (from the base of the lighthouse to the boat) is the 'adjacent' side.
• The tangent function is utilized as \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

By rearranging this equation, we can solve for the 'adjacent' side when the tangent of the angle and the 'opposite' side are known, allowing us to determine the distance to the boat. This relationship gives the tangible application of trigonometry in calculating real-world distances from measured angles.