Applications of Exponents

1. Applications of Exponents Solve real life applications using exponents

2. Simple Interest If a principal of P dollars is borrowed for a period of t years at a per annum interest rate r, expressed as a decimal, the interest I charged is I = Prt

3. Payment period Annually once a year Semiannually twice a year Quarterly four times a year Monthly 12 times a year Daily 365 times a year

4. Compounded Interest When the interest is due at the end of a payment period is added to the principle so that the interest computed at the end of the next payment period is based on this new principle amount (old principle + interest), the interest is said to be compounded. Compound interest is interest paid on principle and previously earned interest

5. Compound interest formula A = accumulated value or future value t = time P = principle r= annual interest rate n = compounded that many times A = P(1+ )nt

6. Example Investing $1000 at an annual rate of 10% compounded annually, semiannually, quarterly monthly and daily will yield what amounts after 1 year?

7. Continuous Compounding The amount A after t years due to a principle P invested at an annual interest rate r compounded continuously A = Pert The amount A that results from investing a principle P of $1000 at an annual rate r of 10% compounded continuously for a time t of 1 year is?

8. Effective rate of interest The effective rate of interest is the equivalent annual simple rate of interest that would yield the same amount as compounding after 1 year.

9. Computing the value of an IRA IRA (individual retirement account) On January 2, 2004 I put $2000 in an IRA that will pay interest 10% per annum compounded continuously. • What will the IRA be worth when I retire in 2035? • What is the effective rate of interest?

10. Present Value Formulas The present value P of A dollars to be received after t years, assuming a per annum interest rate r compounded n times per year is P = A( 1+ )-nt If the interest is compounded continuously then P = Ae-rt

11. Doubling and Tripling Time for an investment How long will it take for an investment to double in value if it earns 5% compounded continuously? How long will it take to triple at this rate?

12. Exponential Growth and Decay Many natural phenomena have been found to follow the law that an amount A varies with time t according to A(t) = A0ekt where A0= A(0) is the original amount (t=0) and k≠0 is a constant. If k>0 then the above equation is said to follow the exponential law or the law of uninhibited growth. If k<0 is said to follow the law of uninhibited decay

13. Uninhibited growth of cells A model that gives the number N of cells in the culture after a time t has passed (in the early stages of growth is) N(t) = N0ekt, k>0 Where N0 = N(0) is the initial number of cells and k is a positive constant that represents the growth rate of the cells

14. Radioactive Decay The amount A of a radioactive material present at time t is given by A(t) = A0ekt k<0 Where A0 is the original amount of radioactive material and k is negative number that represents the rate of decay.

15. Half life All radioactive substances have a specific half life which is the time required for half of the radioactive substance to decay.

16. Newton’s Law of Cooling Newton’s Law of Cooling states that the temperature of a heated object decreases exponentially over time toward the temperature of the surrounding medium The temperature u of a heated object at a given time t can be modeled by the following function u(t) = T +(u0-T)ektk<0 Where T is the constant temperature of the surrounding medium, u0 is the initial temperature of the heated object, and k is a negative constant.