Problem 74 - Contemporary Precalculus

Question

A man walks for 45 minutes at a rate of \(3 \mathrm{mph}\), then jogs for 75 minutes at a rate of \(5 \mathrm{mph},\) then sits and rests for 30 minutes, and finally walks for \(1 \frac{1}{2}\) hours. Find the rule of the function that expresses his distance traveled as a function of time. \([\) Caution: Don't mix up the units of time; use either minutes or hours, not both.]

Step-by-Step Solution

Answer

Answer: D(t) is a piecewise function: - \(\frac{9}{4}\) miles if \(0 \leq t \leq \frac{3}{4}\) hours (walking) - \(\frac{34}{4}\) miles if \(\frac{3}{4} \leq t \leq 1\) hours (jogging) - \(\frac{34}{4}\) miles if \(1 \leq t \leq \frac{3}{2}\) hours (resting) - \(\frac{52}{4}\) miles if \(\frac{3}{2} \leq t \leq 3 \frac{1}{2}\) hours (final walking)

Step 1: Convert time units to hours

First, we need to convert all the given times in minutes into hours. To do this, we'll divide the number of minutes by 60 (since there are 60 minutes in an hour). 45 minutes = \(\frac{45}{60}\) = \(\frac{3}{4}\) hours 75 minutes = \(\frac{75}{60}\) = \(\frac{5}{4}\) hours 30 minutes = \(\frac{30}{60}\) = \(\frac{1}{2}\) hours

Step 2: Calculate distances for each activity

Now, we need to calculate the distance traveled during each activity. The formula to calculate distance is: Distance = Speed × Time - First, walking distance: Distance = 3 mph × \(\frac{3}{4}\) hours = \(\frac{9}{4}\) miles - Second, jogging distance: Distance = 5 mph × \(\frac{5}{4}\) hours = \(\frac{25}{4}\) miles - Third, resting distance: The man is not moving during this time, so the distance traveled during resting is 0 miles. - Fourth, final walking distance: Distance = 3 mph × \(1 \frac{1}{2}\) hours = 3 mph × \(\frac{3}{2}\) hours = \(\frac{9}{2}\) miles

Step 3: Combine distances and express them as a function of time

Now, we will combine the distances traveled during each activity to find a function that describes the distance traveled as a function of time. Let D(t) be the distance traveled (in miles) and t be the time (in hours). D(t) = - \(\frac{9}{4}\) miles if walking - \(\frac{9}{4}\) + \(\frac{25}{4}\) = \(\frac{34}{4}\) miles if jogging - \(\frac{34}{4}\) miles if resting - \(\frac{34}{4}\) + \(\frac{9}{2}\) = \(\frac{34}{4}\) + \(\frac{18}{4}\) = \(\frac{52}{4}\) miles if final walking So, the final function that describes the distance traveled as a function of time is: D(t) = piecewise function: - \(\frac{9}{4}\) if \(0 \leq t \leq \frac{3}{4}\) hours - \(\frac{34}{4}\) if \(\frac{3}{4} \leq t \leq 1\) hours - \(\frac{34}{4}\) if \(1 \leq t \leq \frac{3}{2}\) hours - \(\frac{52}{4}\) if \(\frac{3}{2} \leq t \leq 3 \frac{1}{2}\) hours

Key Concepts

Distance-Time Relationship

Understanding the distance-time relationship is crucial when considering how motion impacts travel over a certain duration. The distance-time relationship involves calculating the distance a person or object covers in a specific amount of time. In the context of this exercise, it is a piecewise function, meaning the distance traveled changes depending on the activity and time interval.

A piecewise function means different formulas are applied at different time intervals for the man walking, jogging, resting, and then walking again. Each of these segments has a distinct rate of motion and contributes to the overall calculation of the total distance.

To accurately model the journey, it's essential to break down each step according to its specific distance and add them cumulatively to find the final distance-time relationship.

Rate of Motion

Rate of motion refers to the speed at which an individual or object moves over time. In this problem, the man’s activities have different rates of motion.

- Walking at 3 mph.
- Jogging at 5 mph.
- Resting, which essentially means a rate of motion of 0 mph.

Each rate of motion influences how the distance accumulates over time, which helps us formulate the piecewise function.**Applying the Rate of Motion**

To find the distance using the rate of motion, we use the formula: \( \text{Distance} = \text{Rate} \times \text{Time} \). This formula is critical in determining how much distance is covered during each part of the journey and is used for each piece of the journey before moving on to the next. By understanding this, we can simplify the calculations and accurately determine the total distance traveled.

Time Conversion

Time conversion is an essential skill when dealing with rates of motion and distance calculations. In most mathematical problems, consistency in units is vital because using different units can lead to confusion and incorrect results.

The original times given in minutes (45 minutes, 75 minutes, and 30 minutes) need to be converted to hours before using them in calculations. This is because the speed is given in miles per hour (mph), so time needs to be consistent.

**Steps to Convert Minutes to Hours**

• Divide the number of minutes by 60 (since there are 60 minutes in an hour). For example:
  \(45 \text{ minutes} = \frac{45}{60} = \frac{3}{4} \text{ hours}\).
• Apply the same process to the rest of the time values in the problem.

Converting time helps ensure that unit measurements align, making the calculations straightforward and accurate.

Mathematical Modeling

Mathematical modeling involves creating equations or functions that represent real-world situations. In this exercise, the piecewise function models the distance traveled by the man. This is done by summing up all the distances from different segments of his activities that were calculated earlier.

**Constructing the Piecewise Function**

• Identify each stage of motion, including walking, jogging, and resting, over the given time period.
• Calculate the distance for each stage using the formula \( \text{Distance} = \text{Speed} \times \text{Time} \).
• Combine these results into a piecewise function.

By breaking down the journey into these stages, mathematical modeling allows us to create a detailed and accurate representation of the man’s travel over the 3.5 hours. This function is powerful as it aids in understanding how variations in speed and activity affect total distance traveled.