28000(1 - .12)x < 8000
Now that you are familiar with several ways to formulate exponential functions that fit given data or a given situation, you can now generate an appropriate equation to fit a specific situation.
Example 1: Let's look at the depreciation of a car. Suppose you purchase a new car for $28,000, and the depreciation rate for the make and model of your car is 12% per year. Now suppose you plan to trade in or sell this car in the future but you need $8,000 for a down payment for your next car. Write an exponential equation that will allow you to determine how many years, after purchase, you can hang on to your car before its value dips below $8,000.
Interactive popup. Assistance may be required. An exponential inequality that would be appropriate for the situation described above would be:
Example 2: In 1995, there were 37,000 manatees in the waters off the coast of Florida. Due to boating accidents and natural events, the manatee population has been decreasing at an annual rate of 8% and is threatening to become an endangered species. Given the exponential function that models this situation, y = 37000(1 - .08)x, write an exponential equation that would be appropriate to find the number of manatees in the waters off the coast of Florida in 2010? (Hint: Remember that the initial year for this data is 1995. Keep that in mind when determining x, the number of time intervals.)
Example 3: Insulin is often used medically to treat some forms of diabetes. When used to regulate sugar in the bloodstream, insulin breaks down by about 4% per minute. The exponential function, A = U(1 - .04)t, will model this situation where U represents the number of units of insulin given a patient and A represents the amount of insulin remaining in the patient's bloodstream after t minutes. Formulate an exponential equation that would allow you to find A, the amount of insulin remaining in a patient’s bloodstream after half an hour, if the patient was initially given 10 units of insulin.
Interactive popup. Assistance may be required.
An exponential equation that would be appropriate for the situation described above would be:
Example 4: In class, you probably discussed the constant e, which is a special irrational number similar to 
. The constant e is referred to as the "natural base" and is often used in applications of exponential functions, including continuously compounded interest. When interest is compounded continuously, we use the exponential function, A = Pert, where A represents the amount in the account after t years, P represents the principal, or the initial investment, and r represents the annual interest rate expressed as a decimal.
Suppose Randy's parents invested $5000 into a college fund for him when he was born and suppose the account receives 3.5% interest compounded continuously. Randy is now 18 years old and is a senior in high school. He wants to go to Texas A&M for college. His family estimates that he will need $16,000 for his freshman year.
Formulate an exponential equation that would allow Randy's family to determine the amount of money in this account after 18 years. Will there be enough money in this college fund for Randy's first year at A&M?
Interactive popup. Assistance may be required. An exponential equation that would be appropriate for the situation described above would be: