Applications of Exponents. Solve real life applications using exponents. 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. Payment period. Annually once a year
Solve real life applications using exponents
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
Annually once a year
Semiannually twice a year
Quarterly four times a year
Monthly 12 times a year
Daily 365 times a year
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
A = accumulated value or future value
t = time P = principle r= annual interest rate
n = compounded that many times
A = P(1+ )nt
Investing $1000 at an annual rate of 10% compounded annually, semiannually, quarterly monthly and daily will yield what amounts after 1 year?
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?
The effective rate of interest is the equivalent annual simple rate of interest that would yield the same amount as compounding after 1 year.
IRA (individual retirement account)
On January 2, 2004 I put $2000 in an IRA that will pay interest 10% per annum compounded continuously.
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
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?
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
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
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.
All radioactive substances have a specific half life which is the time required for half of the radioactive substance to decay.
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.