This section provides two problems to practice solving contextual exponential expressions.

Problem 1:

One common phenomenon that is typically modeled by exponential decay functions is half-life, which is the period of time it takes for one-half of a substance undergoing decay to decay into another substance. 

In 1986, a nuclear reactor accident occurred in Chernobyl (which was then part of the Soviet Union.)  The explosion caused the spread of highly toxic radioactive chemicals, making the region uninhabitable and forcing an evacuation.  If 10 pounds of Plutonium-239, P, were initially released into the atmosphere with the explosion in 1986 and the function  represents the amount of Plutonium-239 that remains after x years, how much of the Plutonium-239 remains in the year 2010? 

Do you think the region will be re-inhabitable in your lifetime?  Why or why not?

Check Your Answer

Enter the given exponential function into Y₁.
Be sure to put the exponent in parentheses.


The question asks how much Plutonium-239 remains in the year 2010, which is 24 years after the explosion. Find the value of this function when x = 24. Use the TABLE features of the calculator to find the functional value when x=24 or evaluate on the HOME SCREEN.



According to our model 9.993 pounds of Plutonium-239 still remain in that area in the year 2010. This makes sense since the half-life of this substance is 24,000 years. In other words, it would take 24,000 years for the original 10 pounds of Plutonium-239 to decay to 5 pounds.

To answer the second question, you need to know that most regulatory agencies, including the EPA, assume that any exposure to radioactive materials carries some risk. Considering that fact and considering that it takes tens of thousands of years for the Plutonium-239 to decay by half, this region will remain uninhabitable well beyond our lifetimes!

Problem 2:

Americans purchase billions of bottles of water each year. The data in the table below shows the number of plastic bottled water sold, in billions, from 1996 to 2006. 

If Americans continue to purchase bottled water at this same rate, what is your prediction for the number of bottles that will be sold in the year 2015?

Check Your Answer

The number of plastic bottles of water sold in 2015 will be approximately 370 billion!

Using a graphing calculator find an exponential function that models the data given in the table. Then, using that model, and either the TABLE feature or the HOME SCREEN, find the functional value when x = 19 since the year 2015 is 19 years after the initial data value, 1996.