Problem 105 - College Algebra with Modeling and Visualization

Question

Running At 2: 00 PM. a runner heads north on a highway, jogging at 10 miles per hour. At 2: 30 PM. a driver heads north on the same highway to pick up the runner. If the car travels at 55 miles per hour, how long will it take the driver to catch the runner?

Step-by-Step Solution

Answer

The driver will take approximately 6.67 minutes to catch the runner.

Step 1: Determine Initial Lead of the Runner

Since the runner starts at 2:00 PM and the driver begins at 2:30 PM, the runner has a 30-minute head start. In those 30 minutes (0.5 hours), the runner covers a distance of \(10 \text{ miles per hour} \times 0.5 \text{ hours} = 5 \text{ miles}\).

Step 2: Calculate the Relative Speed of the Two

The driver's speed is 55 miles per hour and the runner's speed is 10 miles per hour. Therefore, the relative speed at which the driver is closing the distance is \(55 - 10 = 45 \text{ miles per hour}\).

Step 3: Determine the Time to Close the Distance

To catch the runner, the driver needs to close a distance of 5 miles at a speed of 45 miles per hour. The time required is given by \(\frac{\text{distance}}{\text{speed}} = \frac{5}{45}\text{ hours} = \frac{1}{9} \text{ hours} \approx 6.67 \text{ minutes}\).

Key Concepts

Relative Speed

When two objects move towards or away from each other, their relative speed is a crucial factor to consider. This basically means how fast one object is moving compared to the other. Think of it like you're on a conveyor belt at the airport, standing still, and there's another belt going in the opposite direction. The speed at which you move away from someone on that opposite belt is your relative speed.
In our exercise, the runner moves north at 10 miles per hour while the driver, with the same direction, travels at 55 miles per hour. The relative speed, which measures how quickly these two get closer to each other, is found by subtracting the slower speed from the faster one:

• The driver's speed: 55 miles per hour
• The runner's speed: 10 miles per hour
• Relative speed: 55 - 10 = 45 miles per hour

This relative speed shows how quickly the driver shortens the gap between them and the runner. When understanding problems involving two moving objects, especially when they move in the same or opposite directions, calculating the relative speed simplifies the situation.

Distance-Time Problems

Distance-time problems are a common type of question in Algebra that involve calculating one of the variables: distance, speed, or time, given the other two. Generally, they are based on the formula: \[\text{Distance} = \text{Speed} \times \text{Time}\]With this formula in hand, if you know two of the values, you can always find the third.
This problem starts with a runner who has a head start of half an hour, covering a distance of 5 miles at 10 miles per hour. When the driver departs, their task becomes not just traveling at 55 miles per hour but overcoming this 5-mile head-start. We use the formula rearranged to find the time required (Time = Distance/Speed) to close that gap:

• Distance to close: 5 miles
• Relative speed: 45 miles per hour (from previous section)
• Time to catch up: \( \frac{5}{45} \text{ hours} = \frac{1}{9} \text{ hours} \approx 6.67 \text{ minutes} \)

These problems can often stretch across different scenarios, still linked by the core relationship between speed, distance, and time.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world scenario to solve a problem or make predictions. In our scenario, we're essentially converting a real-world situation into equations that can then be solved for the unknowns.
First, we need to accurately define the problem. The runner and the driver moving at different speeds on the same path becomes the model's core. By using key information such as time intervals and speed, we represent these behaviors through mathematical expressions and operations. Here's a simple breakdown of the modeling process for this problem:

• Identify known values: The runner's speed, initial lead, and the driver's speed.
• Decide what needs to be found: The time it takes for the driver to catch up.
• Create equations: Use the relative speed formula and the distance-time formula to bridge over the known data to find the unknowns.
• Calculate the result: In this case, determine how time and speed interplay to overcome the distance gap.

This thoughtful approach transforms word-based questions into mathematical solutions that can be easily solved with equations, showcasing the power and practicality of mathematical modeling.