Problem 16 - Applied Calculus

Question

Table \(1.12\) gives the net sales of The Gap, Inc, which operates nearly 3000 clothing stores. \({ }^{29}\) (a) Find the change in net sales between 2005 and 2008 . (b) Find the average rate of change in net sales between 2005 and 2008. Give units and interpret your answer. (c) From 2003 to 2008 , were there any one-year intervals during which the average rate of change was positive? If so, when? $$ \begin{array}{l} \text { Table } 1.12 \text { Gap net sales, in millions of dollars }\\\ \begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 2003 & 2004 & 2005 & 2006 & 2007 & 2008 \\ \hline \text { Sales } & 15,854 & 16,267 & 16,019 & 15,923 & 15,763 & 14,526 \\\ \hline \end{array} \end{array} $$

Step-by-Step Solution

Answer

(a) $1,493$ million decrease. (b) $498$ million/year decrease. (c) Yes, from 2003 to 2004.

Step 1: Identify the Sales Values for Relevant Years

First, locate the net sales figures for the years 2005 and 2008. According to the table, the sales in 2005 were $16,019$ million and in 2008 were $14,526$ million.

Step 2: Calculate the Change in Net Sales (2005 to 2008)

To find the change in net sales, subtract the sales in 2008 from the sales in 2005: \[ \text{Change in sales} = 16,019 - 14,526 = 1,493 \text{ million dollars}\]

Step 3: Calculate the Average Rate of Change (2005 to 2008)

The average rate of change is found by dividing the change in sales by the number of years over which the change occurred. \[ \text{Average rate of change} = \frac{16,019 - 14,526}{2008 - 2005} = \frac{1,493}{3} \approx 498\text{ million dollars per year}\]

Step 4: Determine One-Year Intervals with Positive Change

Examine each one-year interval from 2003 to 2008 to see where the average rate of change is positive:- 2003 to 2004: \(16,267 - 15,854 = 413\) (positive change)- 2004 to 2005: \(16,019 - 16,267 = -248\) (negative change)- 2005 to 2006: \(15,923 - 16,019 = -96\) (negative change)- 2006 to 2007: \(15,763 - 15,923 = -160\) (negative change)- 2007 to 2008: \(14,526 - 15,763 = -1,237\) (negative change)The interval from 2003 to 2004 had a positive change.

Key Concepts

Understanding Average Rate of Change

The average rate of change is a key concept in applied calculus, especially in analyzing changes over time. It helps us understand how a quantity changes over a specific duration. Think of it like a car's average speed: if you drove 300 miles in 3 hours, the car's average speed was 100 miles per hour.To calculate the average rate of change of net sales at The Gap, Inc. between 2005 and 2008, we need to know the change in sales value and the time period over which this change occurred. The sales in 2005 were \(16,019\) million dollars, and by 2008, they decreased to \(14,526\) million dollars.The formula for the average rate of change is simple:

• Determine the change in sales: \(16,019 - 14,526 = 1,493\)
• Calculate the time interval: \(2008 - 2005 = 3\) years
• Divide the change in sales by the time interval: \(rac{1,493}{3} \approx 498\) million dollars per year

This result means that, on average, The Gap's net sales decreased by \498\ million dollars every year from 2005 to 2008.Interpreting this number offers valuable insights into business performance, indicating trends and informing decision-making processes.

Net Sales Calculation Explained

Calculating net sales involves determining the monetary value of sales made by a company over a period. In assessing The Gap’s performance, we look at their sales carefully and see how these values change. The exercise gives us yearly net sales from 2003 to 2008.Calculating the net sales change between two years needs a simple subtraction:

• Locate the sales for the considered years - here 2005 and 2008.
• Subtract 2008's sales from 2005's sales to find the change: \(16,019 - 14,526 = 1,493\) million dollars.

This shows that the net sales decreased by \1,493\ million dollars over these years. Changes in net sales can inform business strategies and market evaluation.Recognizing whether sales increase or decrease helps managers pivot strategies and seek improvement areas, ensuring the company remains competitive and financially sound.

Analysis of Time Intervals

Analyzing time intervals provides insights into sales trends. It helps evaluate periods of growth or decline, comparing net sales from one interval to another. In this exercise, we examined one-year intervals from 2003 to 2008 to identify where positive sales growth existed:

• 2003 to 2004: Sales increased from 15,854 to 16,267, a positive change of 413 million dollars.
• 2004 to 2005, 2005 to 2006, 2006 to 2007, 2007 to 2008: Each of these intervals demonstrated decreases, with negative changes calculated sequentially as -248, -96, -160, and -1,237 million dollars.

Of all the intervals, the only period illustrating positive sales change was 2003 to 2004. Recognizing these periods aids in understanding when the company was thriving and when it experienced challenges. This analysis supports making informed decisions to boost future performance.