Problem 3 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

According to a study done by Chester Kyle, Ph.D. (Long Distance Cycling, Rodale Press, Emmaus, PA, 1993), adding 6 pounds to a bicycle slowed the rider down by 22 seconds on a certain 2 -mile course. Assume that riding the course without the extra weight took \(K\) seconds (actual time not speci ed). (a) Assuming that this relationship is linear, nd an equation for \(T(w)\), the time needed to complete the course as a function of the amount of extra weight added. (b) What is the rate of change of \(T(w) ?\) Interpret this rate of change in practical terms.

Step-by-Step Solution

Answer

The equation for \(T(w)\) is: \(T(w) = 3.67w + K\). This means that for every extra pound of weight added to the bicycle, the time to complete the 2-mile course increases by approximately 3.67 seconds.

Step 1: Set Up the Initial Condition

The initial condition is when no extra weight is added which takes \(K\) seconds. This can be represented as the point \((0,K)\).

Step 2: Add the Provided Data Point

The problem provides a specific data point: adding 6 pounds of weight leads to an increase in time by 22 seconds. This can be considered a second point: \((6, K + 22)\).

Step 3: Formula for a Linear Function

A linear function is commonly represented as \(f(x) = mx + c\), where \(m\) is the slope (rate of change) and \(c\) is the y-intercept. In this case, \[T(w) = mw + K\]

Step 4: Calculate the Slope

The slope of a line is calculated by the change in y over the change in x. So, \[m = \frac{K + 22 - K}{6 - 0}\] simplifying this gives \(m = \frac{22}{6}\) or approximately 3.67

Step 5: Complete the Equation for \(T(w)\)

Substitute \(m\) into the equation from step 3, we get \[T(w) = 3.67w + K\]

Step 6: Interpret the Slope as Rate of Change of \(T(w)\)

The slope of the line 3.67 represents the rate of change of \(T(w)\). In practical terms, this means that for every extra pound of weight added to the bicycle, the time to complete the course increases by approximately 3.67 seconds.

Key Concepts

Rate of Change

In mathematics, the concept of 'rate of change' provides a way to describe how one quantity changes in relation to another. In this particular scenario, the rate of change refers to how time changes as extra weight gets added to the bicycle.

The rate of change is represented by the slope in a linear equation. It tells us how much the dependent variable (time) increases or decreases for a unit change in the independent variable (weight). Here, the rate of change is calculated by dividing the change in time by the change in weight, resulting in a rate of 3.67 seconds per pound.

This simply means every additional pound of weight results in an increase of 3.67 seconds in the completion time of the cycling course. It's a critical concept that quantifies relationships in linear functions and is particularly useful for understanding how variables interact in practical scenarios.

Problem Solving

Problem solving in mathematics involves a logical approach to understanding and finding solutions to problems. The given exercise is an excellent example.

The task was to find a linear equation that relates time to weight. By breaking down the problem into simpler parts, and following a structured process, we derived the required linear function.

• Identifying variables: Weight as 'w' and Time as 'T'(w).
• Using known points to establish a system: (0, K) and (6, K + 22).
• Applying formulas for linear functions to determine the equation.
• Calculating and interpreting the slope as a practical rate of change.

This strategy not only solves the problem but also helps in understanding the mathematical relationships between variables. Clear steps and logical reasoning are the keys to effective problem solving.

Slope

The slope is a fundamental concept in the world of linear functions and crucial for understanding how variables interact. In a linear equation like \( T(w) = mw + c \), the slope \( m \) describes how steep the line is on a graph, and it also signifies the rate of change.

To find the slope, we calculate the difference in the dependent variable divided by the difference in the independent variable. In the exercise, the slope was calculated as \( m = \frac{22}{6} \), which simplifies to approximately 3.67. This gives us a way to predict changes in time (\( T(w) \)) based on changes in weight. The slope tells us exactly how many seconds are added to the time for each pound of additional weight.

Understanding the slope is not just about calculations; it's about interpreting how two variables relate to each other in a meaningful way.

Weight and Time Relationship

The relationship between weight and time in this exercise demonstrates a direct linear correlation, meaning that changes in one directly affect the other. The equation \( T(w) = 3.67w + K \) shows a straightforward and predictable relationship: as additional weight is added, time increases linearly.

Here's how this works:

• Starting with no added weight, the time is \( K \) seconds.
• As weight increases, time increases according to the slope (3.67 seconds per pound).
• This relationship is represented graphically by a straight line, showing a constant rate of change.

Such relationships are not unusual in real-world scenarios, making linear functions valuable for modeling everyday situations. Recognizing these linear patterns helps us make predictions, optimizing decisions, and understanding the impact of one variable on another.