Problem 7 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Many trainers recommend that at the start of the season, a cyclist should increase his or her weekly mileage by not more than \(15 \%\) each week. (a) If a cyclist maintains a "base" of 50 miles per week during the winter, what is his or her maximum recommended weekly mileage for the fifth week of the season? (b) Find a formula for \(M(w)\), the maximum weekly recommended mileage \(w\) weeks into the season. Assume that initially the cyclist has a base of \(A\) miles per week.

Step-by-Step Solution

Answer

(a) The cyclist's weekly mileage for the fifth week of the season is approximately 101 miles. (b) The general formula for the cyclist's weekly mileage 'w' weeks into the season is \(M(w) = A * (1.15)^w\).

Step 1: Calculate for 5th Week

Start with the base of 50 miles. Each week, add 15% of the previous week's mileage (which is the same as multiplying by 1.15). Since the problem is asking for the 5th week: \(50 * 1.15 * 1.15 * 1.15 * 1.15 * 1.15\). Compute this value.

Step 2: Derive a General Formula

The problem wants a formula for any given number of weeks, with a starting (or 'base') amount of distance 'A'. The 15% increase each week is equivalent to multiplying by 1.15 each week. So after 'w' weeks, the distance is \(A * (1.15)^w\). This can be simplified to \(M(w) = A * (1.15^w)\), where 'M(w)' is the maximum mileage after 'w' weeks.

Key Concepts

Exponential Growth

Exponential growth describes a situation where a quantity increases at a consistent rate over time. This concept is crucial in understanding how repetitive processes can lead to significant changes. In the example of the cyclist's mileage increase, the weekly mileage grows by a consistent percentage, leading to exponential growth.

When something grows exponentially, each amount is a constant percentage more than the last. This means that rather than increasing by the same fixed number each period (like adding 10 every week), it multiplies by the same factor. For the cyclist, this growth is by a factor of 1.15 each week.

Key characteristics of exponential growth include:

• The increase is proportional to the current value. For example, a 15% increase of a previous value will always be a larger amount than that of a smaller base.
• Over time, the growth becomes increasingly rapid because each increase is larger than the one before, due to the accumulating base.

To calculate exponential growth in mileage, we repeatedly multiply the previous week's mileage by 1.15.

Percentage Increase

Percentage increase is a way of expressing the growth of a number by what percent of itself it has grown. In our cyclist example, each week's mileage is calculated by adding 15% of the prior week's distance to itself.

Here's how to visualize it: if the cyclist starts at 50 miles per week, a 15% increase means each week we adjust the base by the percentage of the base. Mathematically, this can be represented as multiplying the base by 1.15.

• A percentage increase turns into multiplying the original amount by 1 plus the percentage increase as a decimal. So, 15% becomes 0.15, and we multiply by 1.15.
• This means each new week's mileage includes the previous week's mileage and an additional 15% of that mileage. This compounding effect is what leads to exponential growth.

Understanding percentage increases helps in planning realistic training schedules and avoiding overtraining.

Mathematical Modeling

Mathematical modeling involves using mathematical expressions to represent real-world situations. In this scenario, we're using an equation to predict the maximum weekly mileage of a cyclist over several weeks.

The given formula, \(M(w) = A \times (1.15)^w\), is a mathematical model. Here, \(M(w)\) represents the mileage, \(A\) is the initial mileage, and \(w\) is the number of weeks. This approach lets riders see their potential progression in a structured way.

Mathematical models provide several benefits:

• They offer a visual representation of how a variable changes over time, which can guide strategic training decisions.
• Models can also estimate future scenarios, which helps to create effective training plans.
• The accuracy of models depends on how well they represent the real-world system, similar to how well our equation fits the cyclist's training goals.

By applying mathematical modeling, cyclists can set goals and track their progress efficiently. This technique also illustrates the power of algebraic functions in real-life applications, bridging the gap between numerical calculations and practical use.