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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

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applications of exponents

Applications of Exponents

Solve real life applications using exponents

simple interest
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
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

compounded interest
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

compound interest formula
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

example
Example

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

continuous compounding
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?

effective rate of interest
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.

computing the value of an ira
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?
present value formulas
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

doubling and tripling time for an investment
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?

exponential growth and decay
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

uninhibited growth of cells
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

radioactive decay
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.

half life
Half life

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
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.