ACME Soap Company Sales Revenue Growth

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Lesson Parent: 
Differentiation V: Derivatives and Rates of Change
Problem: 

The ACME Soap Company has had the following quarterly sales revenue since 2000. Find the average annual rate of sales revenue change. (Note that the quarter is designated by qQY where q is the first, second, third or fourth quarter and Y is the last two digits of the year. So 3Q8 is the third quarter of 2008.)

Quarter Sales Revenue
1Q0 $5,228,017
2Q0 $1,327,016
3Q0 $9,096,589
4Q0 $1,207,787
1Q1 $9,032,373
2Q1 $659,376
3Q1 $391,966
4Q1 $6,156,056
1Q2 $8,664,531
2Q2 $9,429,404
3Q2 $9,333,132
4Q2 $4,748,838
1Q3 $6,855,646
2Q3 $2,978,313
3Q3 $5,187,014
4Q3 $4,164,627
1Q4 $3,072,403
2Q4 $5,909,060
3Q4 $6,955,731
4Q4 $8,581,070
1Q5 $9,709,962
2Q5 $5,823,954
3Q5 $4,630,233
4Q5 $5,001,678
1Q6 $4,392,870
2Q6 $1,177,141
3Q6 $5,907,076
4Q6 $006,286
1Q7 $5,624,043
2Q7 $4,814,293
3Q7 $1,610,801
4Q7 $852,060
1Q8 $6,141,310
2Q8 $707,390
3Q8 $2,059,847
4Q8 $5,173,682
1Q9 $1,366,766
2Q9 $2,451,813
3Q9 $1,329,738
4Q9 $031,297
1Q10 $1,881,218
2Q10 $662,871
3Q10 $4,780,135
4Q10 $8,736,863
1Q11 $3,641,184
2Q11 $9,967,149
3Q11 $2,901,490
4Q11 $6,713,588
Answer: 

SAMPLE SOLUTION

All of our problems have fully worked out, logical solutions. We connect every solution to its underlying lesson so you can understand the concepts needed to solve any problem and reinforce your learning by practicing with similar problems. All of the formulas are available to our subscribers, but here is a sample of the problem solution you will see when you log in:



Solution: 

We purposefully put a lot of data into this problem to make it look harder than it actually is. First, focus in on what you are being asked to do. You are being asked to provide the average rate of change of sales revenue with respect to time. It is with respect to time in years because the problem asks for annual sales revenue change. However, you are being given data quarterly so you need to add the sales revenue for each quarter to get the yearly sales revenue. Once you do that, you can calculate the average rate of change of sales revenue growth. The only difficulty in this problem is the amount of arithmetic that you have to do. This is where a spreadsheet will allow you to calculate everything quickly.

To add more detail let's denote the yearly sales revenue by $R_i$ at year $t_i$. The average rate of change of sales revenue is

<Sign in to see all the formulas>

Let's do a quick example and then provide a table with the yearly average rate of change.

Quarter 2004 2005
1 <Sign in to see all the formulas>9,709,962
2 <Sign in to see all the formulas>5,823,954
3 <Sign in to see all the formulas>4,630,233
4 <Sign in to see all the formulas>5,001,678
Sum <Sign in to see all the formulas>25,165,827
Average Rate of Change
(<Sign in to see all the formulas>\$$647,563 per year

Now here is the full set of yearly data with yearly average rate of change of sales revenue:

Year Yearly Sales Revenue Average Rate of Change
2000 <Sign in to see all the formulas>619,638 per year
2001 <Sign in to see all the formulas>15,936,135 per year
2002 <Sign in to see all the formulas>12,990,307 per year
2003 <Sign in to see all the formulas>5,332,664 per year
2004 <Sign in to see all the formulas>647,563 per year
2005 <Sign in to see all the formulas>13,682,453 per year
2006 <Sign in to see all the formulas>1,417,823 per year
2007 <Sign in to see all the formulas>1,181,032 per year
2008 <Sign in to see all the formulas>8,902,614 per year
2009 <Sign in to see all the formulas>10,881,473 per year
2010 <Sign in to see all the formulas>7,162,324 per year
2011 $23,223,412