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Warm-up Given x = -5, z = 3, a = 4. Evaluate each expression. 2x 25(2)z-1 75(0.5)a+1

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**Applications of Exponential**

Growth & Decay

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**f(x) = a · (b)x a: the starting value b: rate of change (base)**

ModelL ing f(x) = a · (b)x a: the starting value b: rate of change (base) x: variable of time (exponent)

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Problem #1 Fold a piece of notebook paper 50 times & determine the number of layers of paper.

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**Fold a piece of notebook paper 50 times & determine the number of layers of paper.**

Number of Folds 1 2 3 4 # of layers of paper

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**f(x) = a · (b)x a: the starting value b: rate of change (base)**

Modeling f(x) = a · (b)x a: the starting value b: rate of change (base) x: variable of time (exponent) Answer: f(50) = 1 · (2)50 = x 1015 layers

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Problem #2 The Population of the U.S. was 248,718,301 in 1990 and was projected to grow at a rate of about 8% per decade. y = A(1 + r) t Predict the population, to the nearest hundred thousand, for the years 2010 and 2025.

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Solution: Step 1) To obtain the base (rate of change) for exponential growth, add the growth rate to 100%. 100% + 8% = 108% Or 1.08 WHY??

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Step 2) Write the expression for the population x decades after 1990 (since 1990 is your starting year). f(x) = a · (b)x = 248,718,301 · (1.08)x

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**Step 3 a.) Since the year 2010 is 2 decades (20 years) after 1990, substitute 2 for x.**

= 248,718,301 · (1.08)2 = 290,105,026.3 ~290,100,000 people

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**Step 3 b. ) Since the year 2025is 3**

Step 3 b.) Since the year 2025is 3.5 decades (35 years) after 1990, substitute 3.5 for x. = 248,718,301 · (1.08)x = 248,718,301 · (1.08)3.5 = 325,604,866 ~325,600,000 people bonus opp: check the US's current population. Are the predictions on track?

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Problem #3 Elimination The rate at which caffeine is eliminated from the bloodstream of an adult is about 15% per hour. An adult drinks a soda, and the caffeine in his bloodstream usually reaches a peak level of 30 milligrams Predict the amount, to the nearest 10th of a milligram, of caffeine remaining 1 hour after the peak level & 4 hours after the peak level.

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Solution: Step 1) To obtain the base (rate of change) for exponential decay, subtract the rate of decay from 100%. 100% - 15% = 85% Or .85 WHY??

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**Step 2) Write the expression for the caffeine level x hours after the peak level.**

f(x) = a · (b)x = 30 · (0.85)x

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**Step 3 a.) Substitute 1 (1 hour) for x.**

= 30 · (0.85)1 = 25.5 mg

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**bonus opp: how many hours until you're**

Step 3 b.) Substitute 4 (4 hours) for x. = 30 · (0.85)x = 30 · (0.85)4 = 15.7 mg bonus opp: how many hours until you're caffeine free?

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Problem 4: The birth rate for a colony of ants is 60%. The death rate is 46%. Assume that at the beginning of the observations the population was 650. What will the population be in 5 years? 20 years?

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**a)What is the difference between the birth and death rates?**

b)Is the population increasing or decreasing? c) Write an equation that could model this situation. d) What is the population in 5 years and 20 years? f(x) = a · (b)x

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a) 14% b)Increasing c) d) 1252 in 5 years and 8933 in 20 years

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Problem 5: The birth rate for a school of guppies is 56%. The death rate is 64%. Assume that at the beginning of the observations the population was 800. What will the population be in 12 years? 50 years?

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**a) What is the difference between the birth and death rates?**

b) Is the population increasing or decreasing? c) Write an equation that could model this situation. f(x) = a · (b)x d) What will the populations be in 12 and 50 years?

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**f(x) = 800 · (.92)x a) 8% b) decreasing c)**

d) 294 in 12 and 12 in 50 years

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