Problem 52 - College Algebra with Modeling and Visualization

Question

Exercises \(49-54:\) Write a formula for a linear function that models the situation. Choose both an appropriate name and an appropriate variable for the function. State what the input variable represents and the domain of the function. Assume that the domain is an interval of the real numbers. Speed of a Car \(\quad\) A car is traveling at 30 miles per hour, and then it begins to slow down at a constant rate of 6 miles per hour every 4 seconds.

Step-by-Step Solution

Answer

The formula is \( S(t) = 30 - 1.5t \) with \( t \) in the domain \([0, 20]\).

Step 1: Understanding the Problem

The situation involves a car that starts moving at 30 miles per hour and decreases its speed by 6 miles per hour every 4 seconds. This change in speed is constant, indicating a linear relationship.

Step 2: Choosing the Variables

Let the speed of the car be represented by the function \( S(t) \), where \( S \) denotes the speed of the car, and \( t \) represents time in seconds.

Step 3: Determining the Rate of Change

Given the car slows down by 6 miles per hour every 4 seconds, the rate of change in speed is \(-\frac{6}{4} = -1.5\) miles per hour per second.

Step 4: Formulating the Linear Function

The initial speed of the car is 30 miles per hour. Therefore, the linear function for the car's speed is given by \( S(t) = 30 - 1.5t \), where \( t \) is time in seconds.

Step 5: Identifying the Domain

Since the speed cannot be negative, find when \( S(t) = 0 \). Setting the equation \( 30 - 1.5t = 0 \) and solving for \( t \), we get \( t = 20 \). Thus, the domain of the function is \([0, 20]\), where \( t \) represents time in seconds, and the speed remains positive or zero.

Key Concepts

Rate of Change

The rate of change in a linear function tells us how one variable changes in relation to another. In our car speed scenario, it's how quickly the speed of the car decreases over time as it slows down. The concept of rate of change is derived from the basic idea of a slope in mathematics. Here's how we understand it:

• It's the change in the dependent variable (speed, in this case) per unit change in the independent variable (time).
• In our exercise, the car slows at a steady rate of 6 miles per hour every 4 seconds.

To calculate the rate of change, we need to divide the change in speed by the change in time, which in this specific situation is \(-\frac{6}{4} = -1.5\) miles per hour per second. The negative sign here indicates a decrease in speed. Understanding this concept is crucial because it forms the backbone of analyzing how linear functions behave over time, helping us predict future values from known quantities.

Domain of a Function

The domain of a function is all the possible input values (in this case, time) for which the function produces a meaningful output. It is important to identify clear boundaries to ensure that the model we're working with is realistic.In our example of the slowing car, the domain is tied to the period the car remains in motion before stopping. We start with an initial speed, and as time progresses, the speed decreases until it reaches zero. Here’s how the domain is determined:

• We set the linear function, \( S(t) = 30 - 1.5t \), to zero to find when speed stops,\( 30 - 1.5t = 0 \).
• This gives \( t = 20 \), meaning the car comes to a stop after 20 seconds.

The domain is thus \([0, 20]\), including zero at the initial time, where the car is just starting to slow, up to 20 seconds, the point at which it stops.

Modeling Real-World Situations

Modeling with linear functions allows us to depict real-world scenarios in a simplified mathematical form. This helps in making predictions and understanding patterns. Let's break it down using our car speed problem:When we model such a situation, we first identify variables:

• The dependent variable is the car's speed, which we denote as \( S(t) \).
• The independent variable is time, symbolized by \( t \).

Creating the linear function \( S(t) = 30 - 1.5t \) provides a mathematical representation of how the car's speed changes over time. Why is this beneficial?

• It helps predict the car's speed at any future time while the car decelerates.
• Facilitates understanding of the impact of consistent braking over time.

Linear models are powerful because they take complex phenomena and translate them into simple, analyzable forms - valuable for solving many practical, day-to-day problems.