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Section 5.5

Exponential Models

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We will study models of the form:

or

exponential growth

"graph is increasing"

exponential decay

"graph is decreasing"

Notes:

1) b > 0 and is called the "continuous growth/decay rate."

2) a is the "initial value" (the value when x = 0).

3) The parameters a and b are GENERIC. Many models use different letters for the same form of equation. For example,

A = Per t.

The Two Bugs:

Linear Bug

Exponential Bug

Both bugs pass through the points (1,2) and (3,5), but on different paths.

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Procedure for finding an exponential growth/decay model:

Step 1: Use the two data points to create two equations using the given form.

Step 2: Divide the equations and SIMPLIFY using algebra and exponent properties.

Step 3: Solve for b, the growth rate.

Step 4: Use one of the original equations to solve for a, the initial value.

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Example: A quantity is growing according to the model y = a eb x.

When x = 1, y = 75

When x = 3, y = 100.

These are two data points:

(1,75) and (3,100)

(x,y) (x,y)

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divide the

equations

write two equations

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*

*

**

*

(round to 2 places)

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The Model:

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Example 2: "The Petri Dish"

This is where bacteria live.

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(1,150) (4,500)

Write two equations using the two data points.

(t,B) (t,B)

Note: B is in thousands.

B is in thousands

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Divide the equations and simplify.

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The Model:

Use one of the equations and solve for a.

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Example 3

A Population Model

time(in years)

k is the decay rate, since the population is decreasing.

P0 is the

Initial Population

People(in thousands)

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Year 1990 1991 1992 1997 1998 ?t 0 1 2 7 8 ?P 20

Pop.

...Example 3: Finding the population of an island:

In 1992, there were 35,000 people.

In 1997, there were 28,000 people.

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Year 1990 1991 1992 1997 1998 ?t 1 2 7 8 ?P 20

Pop.

0

Fill in the table with the data that you are given.

Note that the variable P represents "thousands of people."

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decay rate

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We must now find the initial value, P0.

*

Note: use the equation from the numerator.

calculator: 28 ÷ e^(-.0446 x 7) =

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P is thousands of people.

t is years since 1990.

P0 = 38.26 meaning 38,260 people were there in 1990.

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point of intersection

t-value ≈ 14

To find when the population is 20,000 people we can estimate using Desmos.

Graph the model and graph the line y = 20 and see where they intersect.

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Finding the time ALGEBRAICALLY when the population will be 20,000:

Set P = 20 and solve for t.

Set P = 20,solve for t.

continued on next page...

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Example 4 -- Exponential Growth Model

t 0P 100 200 400 800 1600

Notice that the time required for P to double is constant(always the same).click here

Definition: The doubling time of an exponential growth function is the interval(time) required for the value to double.

Doubling time is also called "Time to Double."

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Example 5: An investment of $5000 is compounded continuously at 3.5%. How many years will it take for the investment to double in value?

Plug in the given numbers.

Set A equal to twice the initial value.

Solve!

Click here

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Example 6: An investment is compounded continuously for 15 years at 3.5% and yields $1725.36. What was the initial investment?

Plug in the given numbers.

Solve!

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Example 7: An investment of $500 is compounded continuously and doubles in value after 7 years. What was the APR?

Note: "doubles in value after 7 years" means "A = 1000 when t = 7".

Plug in the given numbers.

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...example 7

APR, not the same as r.

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Radioactive decay

Many radioactive substances breakdown(decay) according to a model of the form:

initial quantity

decay rate

The length of time required for half the substance to decay is called the half-life.

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Carbon-14, or 14C is one such substance.

The half-life of Carbon-14 is 5730 years.

Example 8:

Find the decay rate of Carbon-14. Use 6 decimals.

Set Q = half the initial value and t = 5730

Decay rate of 14C.

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Example 9:

A bone fragment contained 12g of carbon-14 when it died and now contains 7g. How long has it been dead?

Note: "g" stands for "gram". It is a unit of mass, which means the amount of material.

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Radium-226, or 226Ra is another such substance.

The half-life of Radium-226 is 1599 years.

Find the decay rate of Radium-226.

Use 6 decimals.

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Plutonium-239, or 239Pu is another such substance.

The half-life of Plutonium-239 is 24,100 years.

Find the decay rate of Plutonium-239.

Use 8 decimals.

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The End.