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Applications of Exponential Functions. 5.3. Objectives: Create and use exponential models for a variety of exponential growth or decay application problems. Example #1 Compound Interest.
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Applications of Exponential Functions 5.3 Objectives: Create and use exponential models for a variety of exponential growth or decay application problems.
Example #1Compound Interest • If you invest $9000 at 4% interest, compounded annually, how much is in the account at the end of 5 years? Use the following formula where A is the final about, P is the initial amount, r is the rate (as a decimal), and t is the time in years.
Example #2Different Compounding Periods • Determine the amount that a $5000 investment over ten years at an annual interest rate of 4.8% is worth for each compounding period. • annually • quarterly This time the formula is slightly modified to include n which is the number of times of compounding.
Example #2Different Compounding Periods • Determine the amount that a $5000 investment over ten years at an annual interest rate of 4.8% is worth for each compounding period. • monthly • daily
Example #3Solving for a Time Period • If $7000 is invested at 5% annual interest, compounded monthly, when will the investment be worth $8500? Solve this problem by graphing as in the previous lesson. Be very careful how this is typed into the calculator It will take about 3.9 years.
Example #4Solving for an Initial Amount • An investment account has a total value of $17,000 after 10 years at 7.4% compounded annually. What was the initial starting value? This equation doesn’t need graphed in order to be solved.
Example #5Continuous Compounding • If you invest $3500 at 3% annual interest compounded continuously, how much is in the account at the end of 4 years? Continuous compounding means that money is added instantaneously when it is made from the interest. For the formula, all variables are the same as before, but e is not a variable, but rather an irrational number programmed into the calculator with an approximate value of 2.71828.
Example #6Population Growth • A newly formed lake is stocked with 900 fish. After 6 months, biologists estimate there are 1710 fish in the lake. Assuming the fish population grows exponentially, how many fish will there be after 25 months? Although it appears that the formula has changed, it really is the same formula as with annually compounded interest. The difference being that a = 1 + r andxis time in months. Since the growth rate is unknown, we must first solve for it in order to predict any future values.
Example #6Population Growth • A newly formed lake is stocked with 900 fish. After 6 months, biologists estimate there are 1710 fish in the lake. Assuming the fish population grows exponentially, how many fish will there be after 25 months? Now that we have solved for a, we can plug it back into the original equation. This time we are solving for the final amount, A, after 25 months (x).
Example #7 • Each day, 15% of the chlorine in a swimming pool evaporates. After how many days will 60% of the chlorine have evaporated? We go back to the first formula for this problem, except since chlorine is evaporating, we subtract r from 1. It might appear that we do not have enough information, but the initial amount and final amount is not necessary to know if percents are given. We assume that the initial amount was 100% or 1.00. If 60% evaporated, then 100% − 60% leaves 40% remaining or 0.40. Now we will solve using a graphing calculator.
Example #7 • Each day, 15% of the chlorine in a swimming pool evaporates. After how many days will 60% of the chlorine have evaporated? Although difficult to read, the intersection takes place with the standard zoom so no other adjustments are necessary. 60% will have evaporated after about 5.6 days.
Example #8Radioactive Decay • The isotope iodine-131 is used to treat some types of thyroid disease. Iodine-131 decays exponentially with a half-life of 8.02 days. If a quantity of iodine-131 is administered to a patient, how long will it take for 99% of it to decay? Like on the previous problem we assume we’re starting with 100%, and subtract 99% from it to obtain the amount remaining (1%). For this formula, x is the time in years and h is the half-life of the substance. A and P remain the same.
Example #8Radioactive Decay • The isotope iodine-131 is used to treat some types of thyroid disease. Iodine-131 decays exponentially with a half-life of 8.02 days. If a quantity of iodine-131 is administered to a patient, how long will it take for 99% of it to decay? The window will need adjusted to fit the intersection on the graph. The view shown required ZOOM 0: ZoomFit and was zoomed out twice. It will take 53.28 days to reach 99% decayed.
