Exponential Growth And Decay Word Problems Answers

Exponential Growth And Decay Word Problems Answers Exponential Growth and Decay Word Problems Answers Explained This document provides detailed solutions and explanations for a variety of word problems involving exponential growth and decay Each problem is categorized by type and includes a stepbystep walkthrough to help you understand the underlying concepts and apply them to realworld scenarios I Exponential Growth 1 Population Growth Problem The population of a town was 5000 in 2010 It has been growing at a rate of 2 per year What will the population be in 2025 Solution Identify the variables Initial population P 5000 Growth rate r 2 002 Time t 2025 2010 15 years Apply the exponential growth formula Pt P1 rt Substitute the values P15 50001 00215 Calculate P15 6719 Answer The population of the town will be approximately 6719 in 2025 2 Investment Growth Problem You invest 1000 in a savings account that earns 4 annual interest compounded continuously How much money will you have after 10 years Solution Identify the variables Principal P 1000 Interest rate r 4 004 Time t 10 years 2 Apply the continuous compounding formula At Pert Substitute the values A10 1000e004 10 Calculate A10 149182 Answer You will have approximately 149182 in the account after 10 years 3 Bacteria Growth Problem A bacterial culture starts with 100 bacteria and doubles every 30 minutes How many bacteria will there be after 3 hours Solution Identify the variables Initial number of bacteria N 100 Doubling time td 30 minutes Total time t 3 hours 180 minutes Calculate the number of doubling periods n ttd 18030 6 Apply the exponential growth formula Nt N 2n Substitute the values N180 100 26 Calculate N180 6400 Answer There will be 6400 bacteria after 3 hours II Exponential Decay 4 Radioactive Decay Problem The halflife of Carbon14 is 5730 years If a sample initially contains 10 grams of Carbon14 how much will remain after 17190 years Solution Identify the variables Initial amount A 10 grams Halflife t 5730 years Time t 17190 years Calculate the number of halflives n tt 171905730 3 Apply the exponential decay formula At A 12n Substitute the values A17190 10 123 Calculate A17190 125 grams Answer After 17190 years 125 grams of Carbon14 will remain 3 5 Drug Concentration Problem A patient is given a dose of medication that decays exponentially with a halflife of 2 hours If the initial concentration is 200 mgL what will the concentration be after 6 hours Solution Identify the variables Initial concentration C 200 mgL Halflife t 2 hours Time t 6 hours Calculate the number of halflives n tt 62 3 Apply the exponential decay formula Ct C 12n Substitute the values C6 200 123 Calculate C6 25 mgL Answer The concentration of the medication will be 25 mgL after 6 hours 6 Value Depreciation Problem A car was purchased for 25000 Its value depreciates at a rate of 10 per year What will its value be after 5 years Solution Identify the variables Initial value V 25000 Depreciation rate r 10 01 Time t 5 years Apply the exponential decay formula Vt V1 rt Substitute the values V5 250001 015 Calculate V5 14580 Answer The car will be worth approximately 14580 after 5 years III Applications and Extensions 7 Compound Interest Problem You invest 5000 in an account that pays 35 annual interest compounded monthly How much will you have after 15 years Solution 4 Identify the variables Principal P 5000 Interest rate r 35 0035 Time t 15 years Number of compounding periods per year n 12 Apply the compound interest formula At P1 rnnt Substitute the values A15 50001 0035121215 Calculate A15 814729 Answer You will have approximately 814729 after 15 years 8 HalfLife Determination Problem A radioactive substance decays to 18 of its original amount in 24 hours What is the halflife of this substance Solution Recognize that 18 represents three halflives 12 12 12 18 Since three halflives occur in 24 hours one halflife is equal to 243 8 hours Answer The halflife of the substance is 8 hours 9 Exponential Growth with Limitations Problem A population of rabbits in a closed environment can grow at a rate of 15 per month However the environment can only support a maximum population of 500 rabbits If the initial population is 50 rabbits model the population growth over time Solution Apply the logistic growth model Pt K 1 KP 1ert Identify the variables Carrying capacity K 500 Initial population P 50 Growth rate r 15 015 Substitute the values and solve for Pt at different time points Answer The model will show that the population initially grows exponentially but eventually levels off as it approaches the carrying capacity Conclusion By understanding the fundamental concepts of exponential growth and decay you can 5 confidently solve realworld problems related to population growth investment returns radioactive decay and many other applications This document provides a strong foundation and clear examples to guide you in applying these concepts to solve a wide range of word problems Remember to identify the key variables select the appropriate formula and carefully substitute the values to arrive at the correct answers