Problem 18 - Applied Calculus

Question

Table 1.17 gives sales of Pepsico, which operates two major businesses: beverages (including Pepsi) and snack foods. \(^{34}\) (a) Find the change in sales between 2003 and 2010 . (b) Find the average rate of change in sales between 2003 and \(2010 .\) Give units and interpret your answer. $$\begin{array}{c|c|c|c|c|c|c|c|c} \hline \text { Year } & 2003 & 2004 & 2005 & 2006 & 2007 & 2008 & 2009 & 2010 \\\ \hline \text { Sakes } & 26,971 & 29,261 & 32,562 & 35,137 & 39,474 & 43,251 & 43,230 & 57,838 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

(a) Change: 30,867 million dollars. (b) Average rate: 4,409.57 million dollars/year.

Step 1: Identify Sales in 2003 and 2010

From the given table, identify the sales figures for the years 2003 and 2010. Sales in 2003 was 26,971 million dollars and in 2010 it was 57,838 million dollars.

Step 2: Calculate the Change in Sales

Subtract the sales in 2003 from the sales in 2010 to determine the change. \[ \text{Change in Sales} = 57,838 - 26,971 = 30,867 \] The change in sales between 2003 and 2010 is 30,867 million dollars.

Step 3: Calculate the Average Rate of Change

To find the average rate of change, divide the change in sales by the number of years between 2003 and 2010. There are 2010 - 2003 = 7 years between 2003 and 2010.\[ \text{Average Rate of Change} = \frac{30,867}{7} = 4,409.57 \] The average rate of change in sales is 4,409.57 million dollars per year.

Step 4: Interpretation of the Average Rate of Change

The average rate of change indicates that, on average, Pepsico's sales increased by approximately 4,410 million dollars each year between 2003 and 2010.

Key Concepts

Change in Sales

Understanding the concept of change in sales is crucial for analyzing a company's performance over time. By looking at sales figures from different periods, we can determine how much a business has grown or shrunk in terms of revenue. In our exercise, we identified the sales numbers for Pepsico in 2003 and 2010. These figures were 26,971 million dollars in 2003 and 57,838 million dollars in 2010. To measure the change, we subtracted the sales figure from 2003 from that of 2010, resulting in a change of 30,867 million dollars.
This calculation shows us that Pepsico's sales increased significantly over this period, indicating a positive growth trajectory. It's important to note that such growth can be influenced by various factors like market demand, economic conditions, and strategic business decisions.
Using change in sales figures helps businesses understand their financial health and plan for future growth opportunities.

Sales Analysis

Sales analysis involves examining sales data to understand market trends, consumer behavior, and overall business performance. This kind of analysis helps businesses make informed decisions and strategies.
In our example, we calculated the average rate of change to see how Pepsico's sales evolved annually between 2003 and 2010. By dividing the total sales change of 30,867 million dollars by the 7-year duration (from 2003 to 2010), we found that the average sales increased by 4,409.57 million dollars per year. This value is significant because it quantifies the sales growth on a yearly basis, providing a clearer view of the company's performance over time.
Sales analysis not only highlights growth but also helps businesses identify periods of exceptional performance or concern. It contributes to strategic decision-making, such as identifying potential investment opportunities or areas in need of improvement.

Pepsico Sales

Analyzing Pepsico's sales gives us valuable insights into the company's business dynamics, especially given its operations in both beverages and snack foods. Between 2003 and 2010, Pepsico showed impressive sales growth from 26,971 to 57,838 million dollars.
This significant increase reflects the strength of Pepsico's brand and its successful market strategies. Factors contributing to such growth can include product innovation, global expansion, and effective marketing campaigns.
Understanding such sales data is crucial for investors, analysts, and business leaders as it indicates the company’s capacity to grow and capture market share. It also plays a role in forecasting future sales and developing competitive strategies to sustain growth. By studying past sales trends, Pepsico can better prepare for future challenges and leverage its strengths to continue thriving in the competitive marketplace.