Problem 49 - Intermediate Algebra

Question

Suppose that Bianca walks at a constant rate of 3 miles per hour. Explain what it means that the distance Bianca walks is a linear function of the time that she walks.

Step-by-Step Solution

Answer

Bianca's distance is a linear function of time because she walks at a constant rate of 3 miles per hour, represented by \( d(t) = 3t \).

Step 1: Understanding Linear Functions

A linear function is a function of the form \( f(t) = mt + b \), where \( m \) is the rate of change, \( t \) is the independent variable, and \( b \) is the y-intercept. Here, \( f(t) \) represents the distance Bianca walks.

Step 2: Identifying Rate of Change

The rate of change \( m \) in Bianca's case is her walking speed, which is 3 miles per hour. This tells us how the distance changes as time increases.

Step 3: Defining the Function

Since Bianca starts walking from a specific point and her speed is constant, the distance \( d \) she walks over time \( t \) hours can be expressed as \( d(t) = 3t \). There is no initial distance when \( t = 0 \), thus \( b = 0 \).

Step 4: Interpreting the Linear Relationship

The function \( d(t) = 3t \) shows a direct proportional relationship between distance and time, indicating that for every hour Bianca walks, the distance increases by 3 miles.

Key Concepts

Rate of Change

The rate of change in a linear function quantifies how one variable changes in relation to another. In the context of Bianca's walk, the rate of change is her walking speed - 3 miles per hour. This tells us that for every hour Bianca walks, the distance she covers increases by 3 miles. Imagine the rate of change as the speedometer in a car; it indicates the speed at which the car is moving. Similarly, for Bianca, this rate captures her constant walking speed.

To recognize the rate of change in a linear function, look for the constant represented by the symbol \( m \) in the function formula \( f(t) = mt + b \). Here, \( m = 3 \), showing the consistent change in distance over time due to Bianca's constant pace.

Independent Variable

In a linear function, the independent variable controls the behavior of the function. It is the input that influences the output. For Bianca, the independent variable is time \( t \), measured in hours. The time Bianca spends walking directly affects the distance she covers. Think of the independent variable as the driver of the function; it’s what keeps the process moving.

Without changes in \( t \), there would be no change in the distance \( d(t) \). This makes time the key factor in determining how far Bianca walks. Whenever you identify a linear function, look for the independent variable, which is often labeled as \( t \) or \( x \), representing the input or control over the function's outcome.

Proportional Relationship

A proportional relationship in mathematics describes a scenario where two quantities maintain a consistent ratio. In Bianca's case, her distance walked over time exemplifies this concept. Every hour she walks results in a consistent addition of 3 miles to her total distance, showing a direct and steady increase.

Proportional relationships appear in functions like \( d(t) = 3t \). The absence of any initial value (\( b = 0 \)) means the relationship starts at the origin; the distance is zero when the time is zero. This demonstrates that distance and time are directly linked, in proportion. Recognizing the straight-line graph of a proportional relationship reinforces the simplicity and predictability of such connections.