Exponential Growth and Decay - PowerPoint Presentation

Slide1

Exponential Growth and Decay

Section 6.4aSlide2

Law of Exponential Change

Suppose we are interested in a quantity

that increases ordecreases at a rate proportional to the amount present…

Can you think of any examples???

If we also know the initial amount of , we can model thissituation with the following initial value problem:

Differential Equation:

Initial Condition:

Note:

k

can be either

positive or negative

What happens in eachof these instances?Slide3

Law of Exponential Change

Let’s solve this differential equation:

Separate variables

Integrate

Exponentiate

Laws of Logs/ExpsSlide4

Law of Exponential Change

Let’s solve this differential equation:

Def. of Abs. Value

Let A = + e

C

–

Apply the

Initial Cond.

Solution:Slide5

Law of Exponential Change

If

y

changes at a rate proportional to the amount present(dy/dt

= ky) and y = y when t = 0, then

0

where

k

> 0 represents growth and k < 0 representsdecay. The number k is the rate constant of theequation.Slide6

Compounding Interest

Suppose that A dollars are invested at a fixed annual interest

rate

r. If interest is added to the account k times a year, the

amount of money present after t years is

0

Interest can be compounded monthly (

k

= 12), weekly (k = 52),daily (k = 365), etc…Slide7

Compounding Interest

What if we compound interest

continuously

at a rate proportionalto the amount in the account?

We have another initial value problem!!!

Differential Equation:

Initial Condition:

Look familiar???

Solution:

Interest paid according to this formula is

compounded

continuously

. The number

r

is the

continuous interest rate

.Slide8

Radioactivity

Radioactive Decay

– the process of a radioactive substance

emitting some of its mass as it changes forms.

Important Point: It has been shown that the rate at which a

radioactive substance decays is approximately proportional tothe number of radioactive nuclei present…

So we can use our

familiar equation!!!

Half-Life

– the time required for half of the radioactive nuclei

present in a sample to decay.Slide9

Guided Practice

Find the solution to the differential equation

dy

/dt = ky

, k aconstant, that satisfies the given conditions.

1. k = – 0.5, y(0) = 200

Solution:Slide10

Guided Practice

Find the solution to the differential equation

dy

/dt = ky

, k aconstant, that satisfies the given conditions.

2. y

(0) = 60, y(10) = 30

Solution:

orSlide11

Guided Practice

Suppose you deposit $800 in an account that pays 6.3% annual

interest. How much will you have 8 years later if the interest is

(a) compounded continuously? (b) compounded quarterly?

(a)

(b)Slide12

Guided Practice

Find the half-life of a radioactive substance with the given decay

equation, and show that the half-life depends only on

k.

Need to solve:

This is

always

the half-life

of a radioactive substance

with rate constant

k

(k

> 0)!!!Slide13

Guided Practice

Scientists who do carbon-14 dating use 5700 years for its half-

life. Find the age of the sample in which 10% of the radioactive

nuclei originally present have decayed.

Half-Life =

The sample is about

866.418

years oldSlide14

Guided Practice

A colony of bacteria is increasing exponentially with time. At the

end of 3 hours there are 10,000 bacteria. At the end of 5 hours

there are 40,000 bacteria. How many bacteria were presentinitially?

There were 1250 bacteria initiallySlide15

Guided Practice

The number of radioactive atoms remaining after

t

days in asample of polonium-210 that starts with y radioactive atoms

is

0

(a) Find the element’s half-life.

days

Half-life =Slide16

Guided Practice

The number of radioactive atoms remaining after

t

days in asample of polonium-210 that starts with y radioactive atoms

is

0

(b) Your sample is no longer useful after 95% of the initial

radioactive atoms have disintegrated. For about how many

days after the sample arrives will you be able to use thesample?

days