Problem 100 - College Algebra and Calculus: An Applied Approach

Question

In 1998, there were 37,107 motor vehicle traffic crashes involving fatalities in the United States. In 2005 , there were 39,189 such crashes. Assume that the trend is linear. Predict the number of crashes with fatalities in 2007. (Source: National Highway Traffic Safety Administration)

Step-by-Step Solution

Answer

The predicted number of crashes involving fatalities in 2007 can be calculated using the linear trend formula. Find the rate of increase in crashes between 1998 and 2005, then use this rate to project the number of crashes in 2007.

Step 1: Calculate the Rate of Increase

First, find the change in number of crashes between 2005 and 1998. This can be done by subtracting the number of crashes in 1998 from the number in 2005. Then divide this difference by the number of years between 1998 and 2005 to find the rate of increase per year. Formula: Rate of Increase = \((\text{Number of crashes in 2005} - \text{Number of crashes in 1998}) / (2005 - 1998)\)

Step 2: Predict the number of crashes for 2007

Next, use the rate of increase determined in Step 1 to predict the number of crashes in 2007. Multiply the rate by the number of years from 2005 to 2007, and add the result to the number of crashes in 2005. Formula: Predicted crashes in 2007 = (\(\text{Number of crashes in 2005}) + (\text{Rate of Increase} \times (2007 - 2005)\)

Key Concepts

Rate of Increase Calculation

Understanding how to calculate the rate of increase is crucial when analyzing trends, especially in a linear fashion. A rate of increase helps us determine how quickly a particular variable grows over a specified period.
To calculate the rate of increase, you need two key data points at different times. In our example of traffic crashes, compare data from 1998 with 2005. First, subtract the number in 1998 from that in 2005. This gives the total increase over the period.
Next, divide this difference by the number of years between the two data points. This division provides the average annual rate of increase. It tells us how much, on average, the number of crashes increased each year.

• Step 1: Find the difference in data points (e.g., crash numbers).
• Step 2: Divide by the number of years between them.

This simple calculation plays a foundational role in predicting future values when assuming a consistent trend.

Traffic Crash Prediction

Predicting future data trends, such as traffic crashes, is vital for planning and safety. A linear trend line offers a method to extend beyond known data points.
The number of crashes in 2005 provides a recent known value. Using the rate of increase, we can predict future values.
Here's a step-by-step look at making such a prediction:

• Use the rate of increase calculated earlier to expand the trend.
• Determine how many years ahead you want to predict, such as two years forward to 2007.
• Multiply the rate of increase by this forward timeframe.
• Add this product to your latest data point from 2005.

This final number estimates what you might expect in the given future year. It's essential to remember that predictions should account for the fact that future conditions could diverge from past trends.

Linear Function Application

The concept of a linear function is a foundation for making predictions based on trends. Linear functions suggest that changes happen at a constant rate over time.
Using a linear function, we assume the relationship between time and another variable, like the number of crashes, remains consistent. This approach is practical when exploring phenomena where other complex influences can be minimally controlled or predicted.
Here's a simple formula for applying a linear function in this type of trend analysis. You use:\[\text{Predicted Value} = \text{Start Value} + (\text{Rate of Change} \times \text{Years Forward})\]This formula helps provide a helpful framework for estimating future developments based on historical linear data. It's widely used for various applications, from economics to environmental studies.