Activity 1: Rational Functions Your company picnic is being held at a state park 15 miles away. There are large differences in how long it took the employees to arrive. Using the equation rate = distance / time, a human resources (HR) representative asks you to finish the table and create a graph of the data to use at an upcoming seminar on safety in the workplace. Question 1: Using the equation r = 15 / t, which of the following is not an ordered pair for this function? A. (0.25 hour, 60 mph) B. (0.30 hour, 50 mph) C. (0.50 hour, 30 mph) D. (0.33 hour, 26 mph) The correct answer to question 1 is letter D: (0.33 hour, 26 mph). This ordered pair does not fit the equation r = 15 / t. Question 2: What will the graph of r = 15 / t look like? A. A straight line B. A parabola C. A hyperbola The correct answer to question 2 is letter C: A hyperbola. Activity 2: Exponential Functions After considering several banks, you select a bank with an interest rate of 3%, compounded weekly. You would like to know how long it will take to double your initial investment. You calculate that it will take about 23 years. The following is the equation: A = P x (1 + r / n) nt The keys to solving this equation is identifying each part of the investment formula and knowing how to solve a logarithmic equation. 1
Question 3: Which of the following correctly represent the variables in the investment formula? A. r = 0.03, n = 2, A = 3P B. r = 3, n = 52, A = 2P C. r = 0.03, n = 52, A = 2P D. r = 1.03, n = 52, A = 2P The correct answer to question 3 is letter C: r = 0.03, t = 52, A = 2P. Question 4: To solve an equation like 2 = 1.00058 52t, what is the next step? A. Square each side of each side of the equation B. Take the log of each side of the equation C. Divide by 1.00058 The correct answer to question 4 is letter B: take the log of each side of the equation. Activity 3: Logarithmic Functions Exponents and logarithms are inversely related. Identify the similar equations based on the relationship between exponential and logarithmic functions. Question 5: What is the equivalent equation to log 2 16 = 4? A. log 4 16 = 2 B. 4 2 = 16 C. 2 4 = 16 D. log100 = 2 The correct answer to question 5 is letter C: 2 4 = 16. 2
Question 6: What is the equivalent equation to log 4 16 = 2? A. log 4 16 = 2 B. 4 2 = 16 C. 2 4 = 16 D. log100 = 2 The correct answer to question 6 is letter B: 4 2 = 16. Question 7: What is the equivalent equation to 10 2 = 100? A. log 4 16 = 2 B. 4 2 = 16 C. 2 4 = 16 D. log100 = 2 The correct answer to question 7 is letter D: log100 = 2. Question 8: What is the equivalent equation to ln e 2 = 2? A. e 1 = e B. e 2 = e 2 C. e = 2 The correct answer to question 8 is letter B: e 2 = e 2. Activity 4: Real-Life Examples After working hard for several years, you were able to save $10,000. You want to invest this money so that in a few years you can use it to put yourself through graduate school. You decided to invest in a certificate of deposit. You want to deposit all of the money you saved ($10,000). You find a bank that offers a special interest rate of 7%. This account compounds continuously. 3
Question 9: You decide that you want to leave the money on the account for 5 years. How much money will you have on your account after 5 years? Reminder: You are investing $10,000 for 5 years at the interest rate of 7%. This account compounds continuously. Which formula would you use? A. A(t) = Pe rt B. A(t) PR / t C. A(t) = PRT D. A(t) = P er T Hint: p = principal r = annual interest rate t = number of years e = 2.718 The correct answer is letter A: A(t) = Pe rt Question 10: You now have the formula that you can use to calculate how much money you will have on your account after a certain number of years. Reminder: You are investing $10,000 for 5 years at the interest rate of 7%. This account compounds continuously. Calculate how much money you will have on your account after 5 years. A(t) = $10,000 x e 0.07 x 5 The result is approximately $14,190.20. 4
Question 11: You decide that the amount you will get after investing the money for 5 years is not enough. Your graduate degree is going to cost no less than $25,000. For how many years do you need to invest your $10,000 to get $25,000? The interest rate and the amount you are investing remain the same. Put the steps in the correct order. A. 25,000 / 10,000 = e 0.07 x t B. $25,000 = $10,000 x e 0.07 x t C. ln ($25,000 / $10,000) = ln e 0.07 x t D. t = [ln ($25,000 / $10,000)] / 0.07 E. ln ($25,000 / $10,000) = 0.07 x t Hint: You use the same equation. However, this time you know A(t) and need to find t. The correct answer is the following: 1. $25,000 = $10,000 x e 0.07 x t 2. 25,000/10,000 = e 0.07 x t 3. ln ($25,000/$10,000) = ln e 0.07 x t 4. ln ($25,000/$10,000) = 0.07 x t 5. t = [ln ($25,000/$10,000)] / 0.07 Use a calculator to calculate how long you will need to invest your money. t = [ln ($25,000/$10,000)] / 0.07 The correct answer is: Approximately 13 years. 5
Question 12: You realize that 13 years is a little too long of a time stretch. You want to start working on your degree in no more than 10 years. You probably need to invest a little more than $10,000. Determine how much you need to invest to get $25,000 in 10 years. A(t) = Pe rt Which of the following equations is correct? A. $25,000 = P x e B. e = $25,000 x p The correct answer is letter A: $25,000 = P x e Question 13: Which of the following equations is correct? A. P = e / $25,000 B. P = $25,000 / e The correct answer is letter B: P = $25,000 / e Question 14: Use your calculator to calculate how much you need to invest. Hint: The equation you need to use is P = $25,000 / e The correct answer is: You need to invest $12,437.80. 6
Question 15: You realize that this is still not what you want. You decide to look for a different bank that can offer you a better annual interest rate. What should your annual interest rate be to receive $25,000 in 10 years if you deposit $10,000? You attempt to calculate your annual interest rate, but after the solution looks wrong, you realize that you have made an error. Which of the following steps contains the error? A. $25,000 = $10,000 x e r x 10 Here, you are simply plugging in values into the formula. B. $25,000 / $10,000 = e r x 10 You are simplifying the equation by isolating like values on one side of the equation. C. $25,000 / $10,000 = ln e r x 10 You are taking the ln of both sides of the equation to bring down the t from the exponent. D. $25,000 / $10,000 = 10 r ln e = 1 E. r = ($25,000 / $10,000) / 10 You are isolating the variable. The correct answer is letter C: $25,000 / $10,000 = ln e r x 10. Step 3 (and consequently 4 and 5) contains the error. You are taking the log of both sides of the equation to bring down the t from the exponent. The correct steps are the following: 3. ln ($25,000 / $10,000) = ln e r x 10 4. ln ($25,000 / $10,000) = 10 r 5. r = ln ($25,000 / $10,000) / 10 Question 16: Use a calculator to find out what your desired interest rate is. r = ln ($25,000 / $10,000) / 10 The correct answer is: Your desired interest rate is approximately 9%. 7