Applications of Logs and Exponentials

Section 3-4

Objectives

I can solve interest rate problems using a calculator

I can solve exponential growth and decay application problems

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Compound Interest

After t years, the balance, A, in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas:

nt

rt

r1. For n compoundings per year: A=P(1+ )

n

2. For continuous compounding: A=Pe

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Compounding Terms (n)

Term Value of (n)Annually 1Semi-annually 2Quarterly 4Monthly 12Daily 365

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Example

Find the accumulated value of an investment of $8000 for 6 years at an interest rate of 6.85% if the money is compounded monthly.

612

12

0685.18000

A

nt

n

rPA

1

720057.18000A51.052,12$A

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Your TurnFind the accumulated value of an investment of

$4800 for 5 years at an interest rate of 3.85% if the money is compounded quarterly.

4 5.0385

4800 14

A

nt

n

rPA

1

204800 1.009625A

$5813.57A

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Example

Find the accumulated value of an investment of $2000 for 8 years at an interest rate of 7% if the money is compounded continuously

Solution:

A= Pert

A = 2000e(.07)(8)

A = 2000 e(.56)

A = 2000 * 1.75A = $3501.35

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Your Turn

rtA Pe(.055)(7)3500A e

Find the accumulated value of an investment of $3500 for 7 years at an interest rate of 5.5% if the money is compounded continuously

$5143.65A

Application Problems

ktA PektA PeGrowth

Decay

A: Final Amount

P: Initial Amount

k: constant for problem

t: time

Application Problems

Many radioactive substances have a decay half-life.

This is the time required for the radioactivity to decay away to ½ its original amount. “k” is the constant

ktA Pe

Application Problems

In almost all application problems, you will have to solve for “k” first

Then you can solve for the required missing information

ApplicationAn ant population in a colony can be

modeled by the equation below where

N is the final number of ants and P is

the intial ant colony size, "t" is the time

in monthsktN Pe

If 10 ants start a colony and there are

20,000 ants after 3 months, how many

ants after 1 year?

320000 10 ke

2.5336k (2.5336)(12)10N e 141.599 10N x

Homework

WS 6-5