1. Section 9C Exponential Modeling (pages 585 – 601)

2. (dependent) = initial value x (1 + r)independent
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 Function grows (or decays) by the same relative amount per unit of time.
Independent variable: t Dependent variable: Q decimal growth rate: r Initial Value: Q0
(dependent) = initial value x (1 + r)independent or
Q = Q0 x (1 + r)t

3. Q = Q0 x (1 + r)t Comments/587 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.

4. Q = 281million x (1.007)100 = 564 million
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 million growth rate: .007 per year independent variable: years since 2000 dependent 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

5. Q = 1.2billion x (.995)50 = .934billion
Ex2/588 China’s one-child policy was originally implemented with the goal of reducing China’s population to 700 million by 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 population b) will this rate of decline be sufficient to meet the original goal?
initial value (in 2000): 1.2 billion rate: per year independent variable: years since 2000 dependent 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.

6. 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 = x (1+.05)t
curvy!

7. 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 = _____

9. Can we solve exactly using the function? Use Properties of Logarithms

10. Can we solve exactly using the function?
Q = x (1+.05)t
30000 = x (1+.05)t
1.25 = (1.05)t
The salary will be $30,000 in 4.6 years

11. 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!

12. When will the acreage be reduced by half?
Q = x (0.93)t
= x (0.93)t
The acreage will be reduced by half in 9.55 years.

13. Units for t and Td (and TH) must be the same.
Other forms for Exponential Functions
Independent variable: t Dependent variable: Q doubling time: Td Initial Value: Q0
Independent variable: t Dependent variable: Q half-life: TH Initial Value: Q0
Units for t and Td (and TH) must be the same.

14. 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.
initial value (at midnight): 20 milligrams half-life: 36 hours independent variable: hours since midnight dependent variable: milligrams of Valium
noon the next day is 12 hours past midnight

15. 10% of its initial value is 10% of 20 mg or .10x20 = 2 mg
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 _________

16. Can we solve exactly using the function?

17. 53/599 Uranium-238 has a half-life of 4. 5 billion years
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?

18. Homework
Pages
#38,#40,#42,#52,#54a