Problem 40 - A Graphical Approach to Precalculus with Limits

Question

A hump-back whale researcher is watching a whale approach directly toward her as she observes from the top of a lighthouse. When she first begins watching, the angle of depression of the whale is \(15^{\circ} 50^{\prime} .\) Just as the whale turns away from the lighthouse, the angle of depression is \(35^{\circ} 40^{\prime} .\) If the height of the lighthouse is 68.7 meters, find the horizontal distance \(x\) traveled by the whale as it approaches the lighthouse.

Step-by-Step Solution

Answer

The whale traveled approximately 146.89 meters.

Step 1: Understand the Situation

The situation involves a right triangle where the observer (researcher) at the top of the lighthouse observes a whale with initial and final depression angles. The height of the lighthouse is the opposite side of the angles, and the horizontal distance is the adjacent side of the angles.

Step 2: Convert Angles into Decimal Form

Convert the given angles from degrees and minutes to decimal form:- Initial angle: \(15^\circ 50' = 15 + \frac{50}{60} = 15.8333^\circ\)- Final angle: \(35^\circ 40' = 35 + \frac{40}{60} = 35.6667^\circ\)

Step 3: Determine Initial Distance using the Initial Angle

Use the tangent of the initial angle to find the initial distance (\(d_1\)):\[tan(15.8333^\circ) = \frac{68.7}{d_1}\]Solving for \(d_1\):\[d_1 = \frac{68.7}{tan(15.8333^\circ)}\]

Step 4: Calculate the Initial Distance

Calculate \(d_1\) using the tangent function:\[d_1 = \frac{68.7}{0.2835} \approx 242.29 \text{ meters}\]

Step 5: Determine Final Distance using the Final Angle

Use the tangent of the final angle to find the final distance (\(d_2\)):\[tan(35.6667^\circ) = \frac{68.7}{d_2}\]Solving for \(d_2\):\[d_2 = \frac{68.7}{tan(35.6667^\circ)}\]

Step 6: Calculate the Final Distance

Calculate \(d_2\) using the tangent function:\[d_2 = \frac{68.7}{0.7202} \approx 95.40 \text{ meters}\]

Step 7: Calculate the Horizontal Distance Traveled

The horizontal distance \(x\) traveled by the whale is the difference between the initial and final distances:\[x = d_1 - d_2 = 242.29 - 95.40 \approx 146.89 \text{ meters}\]

Step 8: Confirm the Calculations

Review each calculation to ensure the angles and trigonometric values were computed correctly. Reconfirm that \(x = 146.89 \text{ meters}\) is accurate by re-checking initial conditions and values used.

Key Concepts

Angles of Depression

Angles of depression may sound complex, but they're quite simple. When you observe something from a higher point, like a lighthouse, you're looking down at an object. The angle your line of sight makes with an imaginary horizontal line from your eyes is the angle of depression.
An example is when the researcher views the whale. She uses this angle to find the distance to the whale, which helps in understanding how far the whale is from the lighthouse.

• The angle of depression is equivalent to the angle of elevation from the whale to the observer, due to alternate interior angles.
• Understanding this angle is crucial to calculating distances in such scenarios.
• In trigonometry problems, these angles are usually given in degrees and sometimes, in minutes.

This exercise helps us see practical applications of measuring angles, especially for determining positions and distances in real life.

Right Triangle

A right triangle is a cornerstone of trigonometry. It has one 90-degree angle. This feature makes the calculations involving other angles and side lengths very predictable using trigonometric ratios.
Let's visualize the scenario with the lighthouse and the whale:

• The lighthouse forms the opposite side of the triangle. Its height is known.
• The distance from the lighthouse base to the whale is the adjacent side.
• The angle of depression helps in calculating these distances, forming a right triangle.

Right triangles are important in solving real-world problems, as they allow easy calculation of unknown sides or angles. They serve as a foundation for understanding many geometric and trigonometric principles.

Tangent Function

The tangent function is vital in solving for distances in geometry, especially with right triangles. Using the tangent ratio is key to the calculations we saw in this exercise.
Here's how tangent works in our problem:

• Tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side, i.e., \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
• By rearranging this formula, we can solve for either side if the angle and one side's length are known.
• In the whale scenario, the tangent function helps us find both the initial and final distances of the whale from the lighthouse.

The tangent function simplifies the problem-solving process, making it easier to find unknown lengths using angles and known lengths. It's an essential tool in trigonometry for practical and theoretical applications.