Exponentials and Logarithms

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Presentation transcript:

Exponentials and Logarithms Applications: Exponentials and Logarithms Objective: Be able to solve problems involving compounding interest. 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 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 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” Annually 1 $2655.00 Semi-Annually 2 $2657.40 Quarterly 4 $2658.64 Monthly 12 $2659.48 Daily 365 $2659.89 Hourly 8,760 $2659.91 Minutely $2659.91 525,600 Continuously $2659.91

Example 1: Find the principal needed to get $2500 after 3 years at 5% compounded monthly? You made $347.56

It will take about 24 years to triple an investment. Example: How long would it take far an investment to triple at a rate of 4.6% compounded quarterly? It will take about 24 years to triple an investment.

Exponential Law Law of Uninhibited Growth/Decay A = Aoekt N(t) = Noekt A0 = Initial Population N0 = Initial Population k = constant k = constant T = Amount of time T = Amount of time A = New population 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?

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? 1st: Find the “k” value 2nd: Find the amount after 7 hours

Example: The half-life of an element is 1710 years. If 15 grams are present now, how much will be present in 40 years? 1st: Find the “k” value 2nd: Find the amount after 40 years

“Compounding Interest” 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? 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? 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? 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?