Applications of Exponential Functions Group Activity 7 Business Project Week #10

Transcription

1 Applications of Exponential Functions Group Activity 7 Business Project Week #10 In the last activity we looked at exponential functions. This week we will look at exponential functions as related to interest rates. In this topic, we give a brief introduction to the math needed for present value and future value problems. If interested in either of these terms, there is plenty of information on the web. Data We will begin by looking at constant percent growth of an investment tool. If one invests $1000 and then checks the account balance every six years, the following amounts are noticed. Year Amount $1000 $1602 $2616 $4230 $6840 Graph your data set on the set of axis below. Examine your graph. What type of regression does it model? Give several reasons for your conclusion.

2 The regression equation Using the regression instructions, find a regression equation that fits your data set. Your regression should be an exponential (ExpReg) regression. When you find your regression write in the form Where = the principal at time t, t = time and = the original investment. In this case, the original investment was $1000. However, your model will give a value slightly less than $1000. If you change the calculated value to 1000, you will be accurate at time zero, but it will make your other answers less accurate. The value of b will represent 1 plus the approximate interest rate of your investment. Hence, the approximate interest rate is b-1. In this example the approximate interest rate is: A further look at interest rates Assume that your money doubles in five years, what would the approximate interest rate be? If your money doubles in five years then. Why? Next we need to solve for b. Use the equation to change our equation to Divide both sides by so that From here, you should be able to use algebra to solve for b. If you need help ask. Remember b = 1 plus the interest rate. Also, when we describe the interest rate, we typically write it as a percent. Thus your interest rate should be 14.9%. Use the process to find interest rates for other additional length of times and complete the chart below. Time to double Interest rate 5 years 14.9% 8 years 10 years 12 years 15 years

3 Next for fun, we will use the same information and multiply time to double by interest rate to obtain a new number. Time to double Interest rate Time to double times interest rate 5 years 14.9% 8 years 10 years 12 years 15 years Notice that the new column is just above 70. If you look on the web for the rule of 72, you will find the reasons why this is. The importance of the rule of 72 is that we can use it to quickly determine the length of time for an investment to double. For example if we know the interest rate is 4% then the doubling time is 72/4= 18 years. We can also figure an approximate interest rate if we know the doubling time. If we desire an investment doubles in ten years, then we need to find an investment that will provide 72/10=7.2% interest rate. What is the approximate time to double if the interest rate is 1. 7% 2. 8% 3. 12% What is the interest rate if the doubling time is 1. 8 years years years Next let s look at the reverse process. Let s borrow $20,000 for a car and making $500 a month payments. After so many months, we check the balance and find months Amount $20,000 $18,457 $16,676 $14820 $12,884

4 First, we generally work interest rate problems with time in years. Rewrite the table changing months into years. years 0 Amount $20,000 $18,457 $16,676 $14820 $12,884 Compute an exponential regression for this data. The equation is: Graph your data and the equation acquired through a regression on the same graph. While the time given in the table does not exceed 2 years, the loan balance will be paid in about 5 years. Make your graph go at least five years. Look at the equation we obtained using an exponential regression. Does it model the behavior of the loan? Why or why not?

5 Consider the following variables A t n i P B Amount of the loan Time (in years) Number of equal payments per year Interest rate based on payment period. This is the annual interest rate divided by n payment Balance In this problem, the number of equal payments per year is 12 and the annual rate is 10%. Including the information we have from above, we have A Amount of the loan $20,000 t Time (in years) t n Number of equal payments per year 12 i Interest rate based on payment period. This is the annual interest rate divided by n (.1/12 which is 10%/12) P payment $500 B Balance B The actual equation used to determine Balance is: ( ) Insert the information and simplify the equation so that the only variables left are B and t.

6 Just checking your work, the answer should be Note: The value for (A P/i) will vary depending on how you rounded the value for the variable i. This equation should leave you questioning why it works. Graph this equation along with the data. This equation does fit the data points and claims that the loan will be paid in approximately 4 years. One of the problems that students have is that they have to accept equations such as this (which by the way is the correct equation) without any understanding of how it is developed. This equation is an exponential equation (notice the variable t is in the exponent). This equation cannot be obtained through a regression analysis. At this time we do not have enough mathematical background to develop this equation. But for those going into business, one should consider what they can do to become better skilled at the equations they need so they can use good critical reasoning skills to make sure the result makes sense. Connections Relating previous skills to a new skill; we plan to provide at least one problem on each test which expands previously learned skills to a new application. In this section, we will be solving exponential equations both graphically and algebraically; as well as look at exponential functions and their inverses. Discuss two ways that you can solve the equation e 9x+5 =14 graphically. Solve the equation in both graphical ways. When you solve them graphically, show where the solution is located on the graph. Now, solve the equation e 9x+5 =14 algebraically. Compare your graphical solutions to your algebraic solution? Which one is more accurate? Explain. Sketch the graph of the function f (x) = e x Sketch the graph of f x. (Remember that the graph of a function and its inverse are reflections over the y = x line.) Write an equation that you think 1 represents f x. (It is important that you sketch the graph accurately and on graph paper if possible. This will allow for a more accurate solution.) 1 Now, algebraically find the equation for f Discuss any similarities and any differences. x. Compare your graphical and your algebraic solution.

7 Exercises Find the equation for the inverse function both graphically by sketching the function and its inverse and algebraically. 1. f x e x 3 2 g t t h t 3 x 4. y log x k n log 2 n r x 2x 1 6 log 3