Section 9C Exponential Modeling (pages 585 – 601)

(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

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.

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

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 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 population b) will this rate of decline be sufficient to meet the original goal? initial value (in 2000): 1.2 billion rate: -.005 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.

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.

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.

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

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 _________

Can we solve exactly using the function?

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?

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