| Using the multi-graph method: | Using the table of values: |
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This section provides exponential equations to practice solving.
Example 1: From 1995 to 2007, the GNP (Gross National Product) in the United States is modeled by the exponential function, y = 7.645(1.05)x, where y is the GNP in trillions of dollars and x is the number of years since 1995. Assuming this model continues to be a good fit for this situation well into the 21st century, find the year when the Gross National Product is expected to be 20 trillion dollars.
Interactive popup. Assistance may be required. Using either graphs or the table of values, the GNP should be 20 trillion dollars when x, the number of years since 1995, is 19.7 or almost 20 years, twenty years after 1995 would be the year 2015.| Using the multi-graph method: | Using the table of values: |
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Example 2: Your family purchased a new laptop computer for $1150. If the laptop’s value decreases by 12% per year, when will the value of the laptop drop below $800?
Interactive popup. Assistance may be required. Formulate an appropriate equation from the contextual situation; the exponential function y = 1150(1 - 0.12)x fits this situation. To find when the value of the laptop will drop below $800 evaluate 800 = 1150(0.88)x or for x, the number of years that will pass. The value of the laptop will depreciate to under $800 in less than 3 years!| Using the multi-graph method: | Using the table of values: |
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Example 3: Insulin is an essential hormone. Without it, the body can’t control its use of glucose, or sugar. Often, doctors prescribe insulin for patients with diabetes. If the insulin breaks down in the bloodstream by about 5% each minute, how much of a 10 unit dosage is still in a patient’s body after half an hour?
Interactive popup. Assistance may be required. The exponential equation y = 10(1 - 0.05)x or y = 10(0.95)x models the given contextual situation. To find the amount of insulin remaining after half an hour substitute 30 in for x, the number of minutes and then evaluate the exponential equation y = 10(0.95)30. This can be done algebraically on the Home Screen by evaluating the right-hand side of the expression or enter the function into Y1 and set TBLSET to 30 and find the amount of insulin remaining in the table. Expect to have a little more than 2 units of insulin remaining after 30 minutes.
Example 4: In 1900, the population of Austin, Texas, was 22,258. From 1900 until 2000, the population has grown exponentially at a rate of 3.6% per year. Use this situation to predict Austin’s population in the year 2020.
Interactive popup. Assistance may be required. Recalling how to formulate an appropriate equation from a contextual situation, the exponential function y = 22,258(1.036)x fits this situation.Solve using algebraic methods:
Begin by putting the exponential function into Y1. Then, go to the home screen to evaluate the function, Y1, for x = 120 since 120 years will have lapsed from 1900 to 2020.
Solve using TABLE features on the calculator:
Begin by putting the exponential function into Y1. Then, go to the table and find the value of Y1when x = 120. Go to TBLSET and change the starting value of your table.
If this exponential model holds true, the population in Austin in the year 2020 will be over one-and-a-half million people or about 1,551,161!