Section 9C Exponential 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).
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
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
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
#38,#40,#42,#52,#54a