Exponential Growth Model Solutions

Hon Discrete 10.3 Exponential Growth Model Worksheet Name: _____ SOLUTIONS _____ Complete on a separate piece of paper. 1) The population of a school increases by 20%, then increases by 10%, then increases by 5%, and then suddenly decreases by 30% in consecutive years. What value does the new student population represent as a percentage of original population of students? (1.20)(1.10)(1.05)(0.70) = 97.02% of original 2) The number of flowers in a botanical garden grow by 15% each year for the next 15 years. If the original number of flowers was 125, then how many flowers are there predicted to be in 10 years? P10 = 125(1.15)10 = 505.69 flowers 3) The number of reported cases of the rare X-virus is to decay by 30% each year for 10 years. There currently are 1,200,000 cases of X-virus worldwide. How cases are expected 6 years from now? P6 = 1,200,000(0.7)6 = 141178.8 cases 4) The number of certain type of bacteria increases at a rate of 20% every year. Suppose there were 288 bacteria in 2009. a. How many bacteria were there in 2007? b. How many bacteria will there be in 2012? 2 288/(1.2) = 200 288*(1.2)3 = 497.664 5) Find P12 for the geometric sequence: 3.5, 9.45, 25.515, 68.8905, … P12 = 3.5(2.7)12 = 525331.2235 6) Find P7 if P0 = 65 and P1 = 48.1. 65(.74)7 = 7.898 7) Find P5 if P8 = 18 and P9 = 72. 18* 4 = 72; P5 = 18/43 = 0.28125 8) Find the given sum using the geometric sum formula. a. 7 + 14 + 28+ 56 + 112 + 224 + 448 + 896 c. 15 Terms: 200 + 180 + 162 + 145.8 + … r = 180/200 = 0.9; 8  2 −1  = 1785 7 − 2 1   b. P0 = 10 and r = 1.2; P0 + P1 + P2 + … + P12  1.2 13 − 1   = 484.97 10  − 1 . 2 1    0.9 15 − 1   = 1588.22 200 0 . 9 1 −   d. P5 = 24 and P6 = 28.8 P4 + P5 + P6 + … + P10 r = 28.8/24 = 1.2; P4 = 20;  1.2 7 − 1   = 258.32 20 1 . 2 1 −   248.80 9) How much interest would you earn on an account with 20% annual interest rate compounded daily that you initially invested $100 into for 1 year? 100(1+.2/365)365 = 122.13 – 100 = $22.13 10) Marsha has an option between two savings accounts. Account #1 she plans to invest $4500 at 20% annual interest rate compounded quarterly and Account #2 she plans to invest $6000 at 15% annual interest rate compounded monthly. Which account should Marsha choose if she only needed to save money for 5 years? 4500(1+.2/4)20 = $11,939.84 6000(1+.15/12)60 = $12,643.09 11) How much money would be in a savings account with 7% annual interest rate compounded monthly that you initially invested $1000 into for 2 year? .07   1000 1 +  12   24 = 1149.81 12) If you initially invested $2500 into for 3 year, how much more money would a savings account with 10% annual interest compounded daily have than a savings account with 10% annual interest rate compounded monthly? .10   2500 1 +  365   3• 365 .10   = 3374.51 and 2500 1 +  12   3•12 = 3370.45 Difference = 4.06 13) How much interest does an account earn if it has an annual interest rate of 8% compounded quarterly when $3000 is invested for 5 years? .08   3000 1 +  4   20 = 4457.84 ; Interest = 1457.84 14) You decide to invest in a savings account with 12% annual interest compounded monthly and will deposit $2000. How much money will the account have after 6 years? How much interest have you made? .12   2000 1 +  12   6*12 = 4094.20 ; Interest = 2094.20 15) You decide to invest $15,000 into an account with 10% annual interest for 5 years. After the 5 years you will remove all the savings and re-invest only the interest from the $15000 into an account with 6% annual interest compounded monthly. After 5 more years how much total money do you have? First 5: 15000(1 + .10 ) = 24157.56 ; Interest = 9157.65 5 .06    Next 5: 9157.65 1 + 12   5*12 = 12352.30 TOTAL = 12352.30 + 15000 = 27,352.30 16) Determine the annual yield if you originally invest $250 and 1 year later the account has $295. 250*r = 295 annual yield = (r -1)*100 = 18% 17) Determine the annual yield if you originally invest $300 and 2 years later the account has $356.43. 300*r2 = 356.43 annual yield = (r -1)*100 = 9% 18) Determine the annual yield of an account with 12% annual interest rate compounded quarterly. (use two decimal places) 4 .12   1 +  = 1.12550881 ; Annual Yield = 12.55% 4   19) Determine the annual yield if after one year an account had $4640 and $4000 was initially invested. 4000r = 4640; r = 1.16 Annual Yield = 16% 20) Determine the annual yield if $63,281.25 was in an account 2 years after a starting deposit of $50,000. 50,000r 2 = 63,281.25; r 2 = 1.265625; r = 1.125 Annual Yield = 12.5% 21) Determine the annual yield of an account with 8% annual interest rate compounded daily. (use two decimal places) .08   1 +  365   365 = 1.08328 ; Annual Yield = 8.33%