Problem 35 - Contemporary Precalculus

Question

A car on a straight road passes under a bridge. Two seconds later an observer on the bridge, 20 feet above the road, notes that the angle of depression to the car is \(7.4^{\circ} .\) How fast (in miles per hour) is the car traveling? [Note: 60 mph is equivalent to \(88 \text { feet/second. }]\)

Step-by-Step Solution

Answer

Answer: To find the speed of the car in miles per hour, follow these steps: 1. Set up a right triangle with the observer on the bridge and the car's positions as its vertices. 2. Use the tangent of the angle of elevation (7.4°) and the height difference (20 feet) to find the horizontal distance between the car's positions. 3. Calculate the speed of the car in feet per second by dividing the horizontal distance by 2 seconds. 4. Convert the speed in feet per second to miles per hour using the conversion factor (1 mile per hour = 88 feet per second). Using these steps, the speed of the car can be calculated in miles per hour.

Step 1: Set up a right triangle

First, draw a right triangle with the observer on the bridge, 20 feet above the road, as one point (A). The other two points are the car's position when it passes under the bridge (B) and the car's position 2 seconds later (C). The angle of depression \(7.4^{\circ}\) (angle BAC) is equal to the angle of elevation (angle ABC).

Step 2: Use trigonometric principles to find the distance the car travels in 2 seconds

In right triangle ABC, we know the angle \(ABC = 7.4^{\circ}\) and side BC = 20 feet. To find side AC, we can use the tangent of the angle: $$ \tan(7.4^{\circ}) = \frac{BC}{AC} $$ Solving for AC: $$ AC = \frac{20}{\tan(7.4^{\circ})} $$ Calculate the value of AC to get the horizontal distance from the car's position when it passes under the bridge to its position 2 seconds later.

Step 3: Convert the distance to speed in feet per second

The distance traveled by the car from point B to point C is equal to the horizontal distance AC. Since the car travels this distance in 2 seconds, we can calculate the speed of the car in feet per second: $$ \text{car speed (ft/s)} = \frac{AC}{2} $$

Step 4: Convert the speed in feet per second to miles per hour

To convert the speed in feet per second to miles per hour, use the following conversion factor: 1 mile per hour = 88 feet per second, therefore, $$ \text{car speed (mph)} = \frac{\text{car speed (ft/s)}}{88} * 60 $$ Calculate the speed of the car in miles per hour using the value found for the car's speed in ft/s. This speed is the final answer.

Key Concepts

Right Triangle

Understanding right triangles is key in solving this problem. A right triangle has one angle of 90 degrees. It consists of a perpendicular height, a base, and a hypotenuse. In this example, the right triangle is formed by the observer's position on the bridge, the car's position under the bridge, and its position two seconds later on the road.
The height of the bridge (20 feet) is one side of the triangle. The horizontal distance the car travels in those two seconds is the base. Meanwhile, the hypotenuse would be the line directly connecting the observer to the car. Visualizing this setup helps to apply trigonometric calculations effectively.

Angle of Depression

The angle of depression is an important concept in this scenario. It is the angle formed between the horizontal line from the observer's eye level and the line of sight down to an object. Here, the angle of depression is noted as 7.4 degrees.
Interestingly, in a right triangle scenario, the angle of depression from the observer's perspective is equal to the angle of elevation from the car's position. This equivalence helps in calculating distances using trigonometric principles. Understanding this angle relationship can guide you in setting up right triangles accurately.

Trigonometric Principles

Trigonometry comes into play in calculating distances from angles. In a right triangle, you can use trigonometric ratios like tangent, sine, and cosine to find unknown sides or angles.
For this problem, the tangent of the angle is particularly useful. The tangent of an angle is the ratio of the opposite side (height of the bridge) to the adjacent side (distance the car travels). Using the formula:
\[ \tan(7.4^{\circ}) = \frac{20}{AC} \]
Solving this equation gives you the value of AC, the distance the car traveled horizontally. Understanding these relationships makes it easier to apply trigonometry in real-world scenarios.

Distance and Speed Conversion

Calculating speed involves understanding how distance relates to time. First, determine the car's speed in feet per second by dividing the horizontal distance by the time (2 seconds).

• Car speed (ft/s) = Distance (ft) / Time (s)

Once you have the speed in feet per second, convert it to miles per hour. This conversion uses the fact that 60 mph equals 88 feet per second.

• 1 mile per hour = 88 feet per second
• Car speed (mph) = (Car speed (ft/s) / 88) * 60

Understanding these conversions is helpful for transitioning between different units, making sense of speed in practical terms.