Problem 26 - College Algebra

Question

Table 3 gives the annual sales (in millions of dollars) of a product from 1996 to 2006 . What was the average rate of change of annual sales (a) between 2001 and 2002 , and (b) between 2001 and 2004\(?\) $$\begin{array}{cccccccc}{\text { Year }} & {1998} & {1999} & {2000} & {2001} & {2002} & {2003} & {2004} & {2004} & {2006} \\ {\text { Sales (millions of dollars) }} & {201} & {219} & {233} & {243} & {249} & {251} & {249} & {243} & {233}\end{array}$$

Step-by-Step Solution

Answer

(a) 6 million dollars per year; (b) 2 million dollars per year.

Step 1: Identify Data Points for 2001 to 2002

From the table, the annual sales in 2001 are 243 million dollars, and the sales for 2002 are 249 million dollars.

Step 2: Calculate Rate of Change from 2001 to 2002

The average rate of change is calculated as the difference in sales divided by the difference in years. For 2001 to 2002, this is: \[ \text{Rate of Change} = \frac{249 - 243}{2002 - 2001} = \frac{6}{1} = 6 \text{ million dollars per year.} \]

Step 3: Identify Data Points for 2001 to 2004

From the table, the annual sales in 2001 are 243 million dollars, and for 2004, the sales are 249 million dollars.

Step 4: Calculate Rate of Change from 2001 to 2004

The rate of change from 2001 to 2004 is calculated as:\[ \text{Rate of Change} = \frac{249 - 243}{2004 - 2001} = \frac{6}{3} = 2 \text{ million dollars per year.} \]

Key Concepts

Annual Sales

Annual sales refer to the total revenue earned from selling goods or services over a one-year period. In this context, it's expressed in millions of dollars and represents how much money a product generates each year. Understanding annual sales is crucial for businesses to assess their performance and make informed decisions moving forward.
For instance, businesses can:

• Evaluate market trends over several years.
• Determine the success of products or marketing strategies.
• Predict future performance and adjust strategies accordingly.

It's easier for businesses to plan for growth and optimize results when they keep track of their annual sales.

Difference in Sales

The difference in sales between two points in time helps us understand how much the sales have increased or decreased. In our example, we find this difference by subtracting the earlier year's sales from the later year's sales.
This step is vital because:

• It shows the magnitude of growth or decline.
• Enables businesses to identify significant changes in performance.
• Helps in evaluating the effectiveness of strategies during certain periods.

In our exercise, calculating the difference between the sales in 2002 and 2001 was crucial to finding the average rate of change. The same principle applies when looking at sales from 2001 to 2004.

Years Interval

The years interval refers to the time between two data points. In our case, it signifies the time difference between years when comparing sales figures from the table.
Calculating the years interval helps in:

• Determining the time span for rate of change analysis.
• Gaining insights into how quickly sales are changing over time.
• Facilitating accurate comparisons of sales data across different periods.

For example, from 2001 to 2002, the interval is one year, while from 2001 to 2004, it extends to three years. This understanding keeps our calculations precise and meaningful.

College Algebra

College Algebra is a branch of mathematics that deals with various algebraic concepts used in higher education. It includes operations with numbers, solving equations, and analyzing functions.
In this exercise, College Algebra principles are applied when determining the average rate of change, which relies on foundational algebra skills.
Understanding average rate of change allows students to:

• Relate algebraic concepts to real-world situations, like business sales.
• Develop skills in analyzing and interpreting data.
• Master techniques for working with equations and inequalities.

By practicing problems like these, learners can enhance their problem-solving skills and gain confidence in using algebra to understand various phenomena.