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Transcript
Objective: 1. Be able to solve problems involving compounding interest. 2. Be able to determine the exponential growth and decay of various populations Critical Vocabulary: Principal, Rate, Compounding, Growth, Decay, Exponential Law, Uninhibited Growth or Decay Simple Interest Formula r A P 1 n nt P = Principal (The amount that you start with) R = Interest Rate (Written in Decimal Form) N = Number of times compounded per year T = Amount of time (in years) John has $2500 that he wants to invest over a period of one year. Fill in the following chart based on the interest rate of 6.2%. Compounded # of Times Formula Value of “A” 1 $2655.00 2 1 .062 A 25001 1 Semi-Annually 2 $2657.40 Quarterly 4 .062 A 25001 2 4 .062 A 25001 4 12 .062 A 25001 12 12 $2659.48 Annually Monthly Daily Hourly 365 $2659.89 8760 $2659.91 525600 $2659.91 365 .062 A 25001 365 8,760 .062 A 25001 8760 Minutely 525,600 Continuously A Pe rt .062 A 25001 525600 $2658.64 A 2500e.062 $2659.91 Example 1: Find the principal needed to get $2500 after 3 years at 5% compounded monthly? r A P 1 n nt .05 2500 P1 12 P P $2152.44 36 2500 .05 1 12 36 You made $347.56 Example: How long would it take far an investment to triple at a rate of 4.6% compounded quarterly? r A P 1 n nt .046 300 1001 4 4t 4t .046 log 3 log 1 4 .046 log 3 4t log 1 4 log 3 t .046 4 log 1 4 t 24.01994067 It will take about 24 years to triple an investment. Exponential Law A = Aoekt A0 = Initial Population k = constant T = Amount of time A = New population Law of Uninhibited Growth/Decay N(t) = Noekt N0 = Initial Population k = constant T = Amount of time N(t) = New population Example: The growth of an insect population obeys the equation A = 700e0.07t where t represents the number of days. After how many days will the population reach 3000 insects? 3000 700e 0.07t 30 e 0.07t 7 30 ln ln e 0.07t 7 30 ln 0.07t ln e 7 30 ln 0.07t 7 30 ln 7 t 0.07 t 20.78981761 days Example: A culture of bacteria obeys the law of uninhibited growth. If there are 800 bacteria present initially, and there are 1100 present after 2 hours, how many will be present after 7 hours? N (t ) N 0 e kt 1st: Find the “k” value 1100 800e 2 k 11 e 2k 8 ln N (t ) N 0 e N (t ) 800e (.159)( 7 ) 11 ln e 2 k 8 N (t ) 2434.78011 Bacteria 11 2k ln e 8 11 ln 2k 8 ln ln 11 8 k 2 2nd: Find the amount after 7 kt hours k .159 Example: The half-life of an element is 1710 years. If 15 grams are present now, how much will be present in 40 years? N (t ) N 0 e kt 1st: Find the “k” value 2nd: Find the amount after 40 years 15 15e1710k 2 N (t ) N 0 e kt N (t ) 15e( .000405)( 40) 1 e1710k 2 ln 1 ln e1710k 2 ln 1 1710k ln e 2 1 1710k 2 1 ln 2 k 1710 N (t ) 14.75895771 grams ln k .000405 “Compounding Interest” 1. 2. 3. Joe wants to invest $3,000.00 in a CD (Certificate of Deposit) for 1 year. His bank is offering to compound the interest monthly at a rate of 4.23%. How much will he have when the CD matures? Andy invests $2,700.00 in a CD at an interest rate of 4.6% for 9 months. If the interest gets compounded continuously, how much will he have at the end of the term? How many years will it take for an initial investment of $7,000.00 to grow to $9,500.00 at a rate of 6% compounded quarterly? 4. How many years will it take for an investment to triple if it is invested at 7.4% per annum compounded monthly? What if it were compounded continuously? 5. In three years you want to purchase a TV that costs $1200. The bank is currently offering an interest rate of 5.25% compounded daily. How much should your initial investment be so you can buy the TV in three years? 6. How long will it take for $1,300 to turn into $5,000 at and interest rate of 6.7% per annum compounded semiannually? What if it were compounded continuously?