1 / 19

Section 9C Exponential Modeling (pages 585 – 601)

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).

vian
Télécharger la présentation

Section 9C Exponential Modeling (pages 585 – 601)

An Image/Link below is provided (as is) to download presentation 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. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Section 9CExponential Modeling(pages 585 – 601)

  2. 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

  3. 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.

  4. 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

  5. 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.

  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 = 24000 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 = _____

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

  9. 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

  10. 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!

  11. 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.

  12. 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.

  13. 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

  14. 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 _________

  15. Can we solve exactly using the function?

  16. 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?

  17. Homework Pages 598-599 #38,#40,#42,#52,#54a

More Related