Problem 35 - College Algebra with Modeling and Visualization

Question

A driver's distance \(D\) in miles from a rest stop after \(x\) hours is given by \(D(x)=75 x\) (a) How far is the driver from the rest stop after 2 hours? (b) Find the slope of the graph of \(D .\) Interpret this slope as a rate of change.

Step-by-Step Solution

Answer

(a) 150 miles. (b) 75 miles per hour as a rate of change.

Step 1: Identify the Formula

The formula given for the distance \(D(x)\) from the rest stop is \(D(x) = 75x\). This means the distance depends linearly on the time \(x\) in hours.

Step 2: Calculate Distance for 2 Hours

Plug \(x=2\) hours into the formula to find the distance. \[D(2) = 75 \times 2 = 150\] miles.

Step 3: Determine the Slope

In the equation \(D(x) = 75x\), the term 75 represents the slope of the linear function.

Step 4: Interpret the Slope

The slope of 75 indicates that the rate of change of distance with respect to time is 75 miles per hour. This means the driver travels 75 miles every hour.

Key Concepts

Rate of Change

When we talk about the "rate of change" in the context of a linear function, we refer to how much one quantity changes in response to a change in another quantity. Here, we are looking at how the distance changes over time. With the given function, the rate of change is directly linked to the slope of the equation.For the driver's distance problem, the function is given by: \(D(x) = 75x \)This means that for each hour (\(x\)) the distance (\(D\)) increases by 75 miles. This 75 is essentially our rate of change. It's a consistent number that tells us how fast the driver is moving away from the rest stop for each hour they drive. In simpler terms, the driver travels at a constant speed, and this speed is represented by the rate of change of 75 miles per hour.In any linear scenario, the rate of change is crucial because it helps to understand how quickly or slowly things are happening over time.

Slope Interpretation

Understanding the slope of a linear function is pivotal as it gives us a visual and numerical grasp of how the relationship between two variables behaves. In the driver's distance problem, the slope is essentially the number that represents how steeply the line of the graph increases as time progresses.The equation given is: \(D(x) = 75x \)Here, 75 is the slope of the function. But what does the "slope" actually mean in real life? Well, the slope represents the speed or the pace at which we move away from the rest stop.

• A positive slope like 75 means the driver is moving away consistently, increasing their distance from the rest stop every hour.
• If instead the slope were negative, it would indicate moving closer back towards the rest stop.
• A slope of zero would mean the driver isn't moving at all; they'd stay at the same distance from the rest stop regardless of the time that passes.

By interpreting slopes properly, one can predict behaviors and trends for various situations.

Distance-Time Relationship

Drawing the connection between distance and time is easier with a distance-time graph, which visually represents how far someone travels over time. In linear functions like the one given in the exercise, this relationship is straightforward as it forms a straight line.In the equation \(D(x) = 75x\), every increase in time (\(x\)) of one hour causes an increase in distance (\(D\)) of 75 miles. This constant relationship gives us a direct proportionality between time and distance, where increasing time results in a consistent increase in distance.

• This linear relationship is represented by a straight line on a graph, where the slope (75 in this case) shows the steepness of the line.
• The steeper the line, the faster the rate of movement or speed, meaning the driver covers more distance in a shorter time.
• The y-intercept of this function is 0, which means that at time 0 (or when the driver hasn't started driving), the distance is also 0.

Essentially, understanding a distance-time relationship helps predict future positions and understand past movements based on the time driven.