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Section 9C Exponential Modeling (pages 585 – 601)

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Section 9CExponential Modeling(pages 585 – 601)

Exponential Growth (Decay)occurs when a quantity increases (decreases) by the same relative amount—that is, by the same percentage—in each unit of time.(Powertown: -- 5% each year, investments)

A Exponential Functiongrows (or decays) by the same relative amount per unit of time.

Independent variable: tDependent variable: Qdecimal growth rate: rInitial Value: Q0

(dependent) = initial value x (1 + r)independent

or

Q = Q0 x (1 + r)t

Comments/587

- Q = Q0 x (1 + r)t
- Units for r and for t must be the same.
- If Q0 is the initial value at time t0, then t must be measured in units since t0.
- In this formula, r is the decimal form of the percentage growth/decay rate
- If r > 0, then quantity grows exponentially. If r < 0, then quantity decays exponentially.

Ex1/587 The 2000 census found a US population of about 281 million. a) Write a function for the US population that assumes a exponential growth at 0.7% per year. b) Use the function to predict the US population in 2100.

initial value (in 2000): 281 milliongrowth rate: .007 per yearindependent variable: years since 2000dependent variable: population

Q = 281million x (1+.007)t

year 2100 is 100 years since 2000, so t = 100

Q = 281million x (1.007)100 = 564 million

Ex2/588 China’s one-child policy was originally implemented with the goal of reducing China’s population to 700 million by 2050. China’s 2000 population was about 1.2 billion. Suppose China’s population declines at a rate of 0.5% per year. a) write a function for the exponential decay of the populationb) will this rate of decline be sufficient to meet the original goal?

initial value (in 2000): 1.2 billionrate: -.005 per yearindependent variable: years since 2000dependent variable: population

Q = 1.2billion x (1-.005)t

year 2050 is 50 years since 2000, so t = 50

Q = 1.2billion x (.995)50 = .934billion

With this model, the predicted population in 2050 is 934,000,000 and so the goal of 700,000,000 will not be met.

What do graphs look like?

43/599 Your starting salary at a new job is $2000 per month and you get annual raises of 5% per year.a) create an exponential function.b) create a table showing Q values for the first 15 units of time.c) make a graph of the exponential function.

Q = 24000 x (1+.05)t

curvy!

Using the graph

43/599 Your starting salary at a new job is $2000 per month and you get annual raises of 5% per year.Using the graph determine:a) your salary after 12 years.b) when your salary will be $30000.

When t = 12, Q = _______.

Q = $30,000 when t = _____

Can we solve exactly using the function?

Use Properties of Logarithms

Can we solve exactly using the function?

Q = 24000 x (1+.05)t

30000 = 24000 x (1+.05)t

1.25 = (1.05)t

The salary will be $30,000 in 4.6 years

What do graphs look like?

39/598 A privately owned forest that had 1 million acres of old growth is being clear cut at a rate of 7% per year.a) create an exponential function.b) create a table showing Q values for the first 15 units of time.c) make a graph of the exponential function.

Q = 1million x (1-.07)t = 1mill.x (.93)t

curvy!

When will the acreage be reduced by half?

Q = 1000000 x (0.93)t

500000 = 1000000 x (0.93)t

The acreage will be reduced by half in 9.55 years.

Other forms for Exponential Functions

Independent variable: tDependent variable: Qdoubling time: TdInitial Value: Q0

Independent variable: tDependent variable: Qhalf-life: THInitial Value: Q0

Units for t and Td (and TH) must be the same.

51/599 The drug Valium is eliminated from the bloodstream exponentially with a half-life of 36 hours. Suppose that a patient receives an initial dose of 20 milligrams of Valium at midnight.a) [function] How much Valium is in the patient’s blood at noon the next day?b) [graph] Estimate when the Valium concentration will reach 10% of its initial level.

initial value (at midnight): 20 milligramshalf-life: 36 hoursindependent variable: hours since midnightdependent variable: milligrams of Valium

noon the next day is 12 hours past midnight

51/599 The drug Valium is eliminated from the bloodstream exponentially with a half-life of 36 hours. Suppose that a patient receives an initial dose of 20 milligrams of Valium at midnight.a) [function] How much Valium is in the patient’s blood at noon the next day?b) [graph] Estimate when the Valium concentration will reach 10% of its initial level.

10% of its initial value is 10% of 20 mg or .10x20 = 2 mg

Q is 2 mg when t is about _________

Can we solve exactly using the function?

53/599 Uranium-238 has a half-life of 4.5 billion years.You find a rock containing a mixture of uranium-238 and lead. You determine that 85% of the original uranium-238 remains; the other 15% decayed into lead. How old is the rock?

Homework

Pages 598-599

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