This section uses real-world examples to model contextual situations involving exponential equations. A graphing calculator is needed.

Example 1:

The demands on the IT Departments of corporations continues to increase because the capacity to store data must keep pace with the growing volumes of data as well as the demand for real-time access to the stored information.

Some examples of institutions continually increasing data storage are:

• The U.S. Library of Congress - almost 160 terabytes of data as of February, 2010, according to Wikipedia
• The banking industry - they must keep records of customers' checks, ATM transactions, and credit card information
• The healthcare industry - they must keep medical and pharmaceutical records.

The volume of data that requires storage might become problematic in the future.

Records show that, in 1998, there were 40.5 terrabytes* (*A terabyte is equivalent to 1000 gigabytes or 10^12 or 10,000,000,000,000 bytes of information.) of data stored in data warehouses.
The exponential function y = 40.5 (1 + .28)^x can be used to model this growth since 1998.

Use this model to predict the amount of data storage needed in the year 2015.

Check Your Answer

Using the TABLE features on the calculator, enter the function y=40.5 (1+28)^x into Y₁ and then look in the table to find when X is 17 (the year 2015 is 17 years after 1998).

Or, go to the Home Screen and evaluate the function when x = 17.

If this model holds true, the amount of data storage necessary in 2015 will be 2,692 terrabytes!

That's a LOT of data!

Example 2:

The table below lists women's and men's median earnings, in dollars, from 1984 through 2004, as reported by the U.S. Census Bureau. No doubt that the gap between men's earnings and women's earnings is narrowing. Will women's earnings overtake men's?

Find exponential functions that model the growth of men's median earnings and women's median earnings and then use a graphing calculator to decide whether or not women's median earnings will exceed men's, and if so, in what year will it occur?

Check Your Answer
Enter the data from the table into L₁, L₂ and L₃, to create a scatterplot. Perform a regression analysis to find an exponential model for Men's and for Women's earnings. Change the window to accommodate viewing their respective earnings over an extended period of time. Then, using features on the calculator, find the intersection of the two functions. According to these models, women's earnings will overtake men's earnings in about 45 or 46 years after 1984 or approximately in the year 2029 or 2030!

Example 3:

The table below shows the amount of a 20 mg dose of medication remaining in a person's bloodstream over eight hours. If 2.0 mg. is the therapeutic dosage and if the patient takes this medicine twice a day, will the amount of medication in the bloodstream dip below the therapeutic level in between doses? Why or why not?

Need a Hint?
Find an exponential model that fits the data in the table to see how much time lapses before the level of medication is below 2.0 mg.

Check Your Answer

Enter the data from the table into L₁ and L₂ to create a scatterplot, and perform a regression analysis to find an exponential model for the medication remaining in the person's blood stream over time.

You will need to change the window to accommodate viewing the graph.

Using the Multi-Graph feature on the calculator with
Y₁ = 20 (0.84)^x and y₂ = 2.0, find the intersection of the two functions.

The intersection point is at (13.272,2) which means that the medication level in the person's bloodstream will dip below the therapeutic dosage a little more than 13 hours and 15 minutes after taking the medication. If the person takes the medicine twice a day, 12 hours apart, then the level of medication in the bloodstream will not dip below the therapeutic level of 2.0 mg.