Lesson:

What's New Read and analyze the problem below to answer the questions that follow. What a Surprise! Today is Alexa's birthday. Her parents want to give her a surprise, it is a savings account passbook with her name as the account holder. Her parents deposited an amount of P20,000.00 on the account at the time she was born. They think that it is about time for Alexa to manage her account. If you were Alexa, what would be your reaction if the passbook will be given to you as a birthday gift? Now that she is 18 years old, how much money will be in her savings account, if the money was invested with an interest of 3% compounded quarterly since the time it was deposited? What is It The previous activity is an example of real-life situations involving exponential function. If you doubt your answer or if you do not know what you are going to do to answer the problem above, that's okay. Well, you may read first the examples here, and you may go back to the activity after you fully understand how to solve involving compound interest. The following are the applications of exponential functions, equations, and inequalities to real-life problems.Real-Life Problems Involving Exponential Function Compound Interest Example 1: Danielle deposited P5,000.00 in an account that offers 6% interest compounded semi-annually. How much money is in his account at the end of three years? The formula for compound interest is A = P (1 + ")" where A = final amount, P = principal or the initial amount, " = interest rate, n = number of times interest is compounded in one year, t = number of years. Solution: Given: P = 5000 r = 6% or 0.06 n = 2 (semi - annually) t = 3 Find A. 0.06 2(3) A = 5000 (1+-2 A = 5000 (1 + 0.03)" A = 5000(1.03) = 5000(1.194) = 5970.26 Therefore, after three years, the amount of money in Danielle's account is P5,970.26 Note: If interest is compounded annually, n = 1. If interest is compounded semi-annually, n = 2. If interest is compounded quarterly, n = 4. If interest is compounded monthly, n = 12. Looking at this example, I believe that you are now ready to check your answer on the What's a Surprise Problem. Do you think you got it right? I believe you do. Population Growth and Decay In the module entitled Representing Real-Life Situations Using Exponential Functions, you encountered problems like population growth and decay. This time, you will encounter the population once again but with the concept of the natural exponential function. The natural exponential function is the function /(x) = e*. (If you want to know about this number, you can read the book "e: The Story of a Number", by Eli Maor.) Example 2: A certain bacterium, given favorable growth conditions, grow continuously at a rate of 5.4% a day. Find the bacterial population after twenty-four hours, if the initial population was 500 bacteria. When you read a problem that suggests growth continuously, you should be thinking "continuously-compounded growth formula". For this situation, the formula is A = Poetwhere A = population after a certain period Po - initial population r = rate of change (growth rate but sometimes it is called decay rate) t = time (growth/decay rates in contexts might be measured in minutes, hours, days, etc.) Solution: Given: P = 500 r = 5.4% or 0.054 t = 1 day Find A. Note: 24 hours is converted to 1 day because the growth rate was expressed in terms of a given percentage per day. Thus, A = 500g0.054(1) A = 527.74 Therefore, there will be about 528 bacteria after twenty-four hours. Real-Problems involving Exponential Equation and Inequalities Exponential equations and inequalities are equations and inequalities in which one or both sides involve a variable exponent. They are useful in situations involving repeated multiplication, especially when being compared to a constant value, such as in the case of interest. For instance, exponential inequalities can be used to determine how long it will take to double one's money based on a certain rate of interest. Example 3: Suppose that a population of a colony of bacteria increases exponentially, at the start of the experiment, there are 1000 bacteria. One hour later, the population has increased to 1200 bacteria. How long will it take for the population to reach 5000 bacteria? Round your answer to the nearest hour. Solution. Given: A = 6000 P = 1000 200 10.2 1000 Find t. 6000 = 1000e(02) - (This is an exponential equation) 6000 1000e(0.2) 1000 1000 Multiplication Property of Equality 6 = e(0.2) In 6 = In e (0.2) Changing exponential to logarithm In 6 = 0.2t Property of logarithm In 6 0.2 t = 8.96 Therefore, it will take 8.96 hours to reach 5000 bacteria.Example 4: Michael owns P15,000.00 and he wants to invest his money into an account that will double his money. He is thinking of a financial institution that can make his dream come true. He is considering to invest his money in a lending company which offers a 15% interest compounded quarterly. For how long, will he invest his money in that company to earn at least twice as much as he has now? Given: A 2 2(15000) (to earn at least twice as much as he has now) P = 15000 r = 15% or 0.15 n = 4 Find t. Why? 2(15000) 2 15000(1 + 15)+ (This is an exponential inequality) 3000015000 2 (1 + 0.0375)4 Simplify 30000 - 15000(1.0375)4 15000 15000 Multiplication Property of Equality 2 2 (1.0375)# 4t 2 log1.0375 2 Changing exponential to the logarithm log 2 log 1.0375 = 18.83 Change-of-base formula log, x = 1082 At 2 18.83 Substitution t 2 4.71 Multiplication Property of Equality Therefore, after at least 4.71 years Michael's money will be P30,000.00 What's More Analyze the given problem and answer the questions that follow: Activity 1.1 "I - Predict Mo" 1. In 2015, a certain municipality in Quezon Province has a population of 45, 300. Each year, the population increases at a rate of about 5%. a. What is the growth factor of the municipality? b. Determine an equation to represent the problem. c. What is the population of the municipality in 2020? Use the equation in letter (b]. d. If the population continues to increase at the same rate, what is the population in 2025. 2. In the early stages of the COVID-19 epidemic in the Philippines, there were 50 persons infected but each day the number rose by 5%. After how many days would about 300 persons be infected