§ 9.6 Exponential Growth and Decay; Modeling Data

Exponential Growth One of algebra’s many applications is to predict the behavior of variables. This can be done with exponential growth and decay models. With exponential growth or decay, quantities grow or decay at a rate directly proportional to their size. Populations that are growing exponentially grow extremely rapidly as they get larger because there are more adults to have offspring. For example, the growth rate for world population is approximately 1.2% or .012. This means that each year world population is 1.2% more than what it was in the previous year. In 2007, world population reached 6.6 billion. Blitzer, Intermediate Algebra, 5e – Slide #2 Section 9.6

Exponential Growth Model Exponential Growth & Decay Exponential Growth Model The mathematical model for exponential growth is given by If k > 0, the function models the amount, or size, of a growing entity. is the original amount, or size, of the growing entity at time t = 0, A is the amount at time t, and k is a constant representing the growth rate. y t Blitzer, Intermediate Algebra, 5e – Slide #3 Section 9.6

Exponential Decay Model Exponential Growth & Decay Exponential Decay Model The mathematical model for exponential decay is given by If k < 0, the function models the amount, or size, of a decaying entity. is the original amount, or size, of the decaying entity at time t = 0, A is the amount at time t, and k is a constant representing the decay rate. y t Blitzer, Intermediate Algebra, 5e – Slide #4 Section 9.6

Exponential Growth EXAMPLE (a) In 2000, the population of the Palestinians in the West Bank, Gaza Strip, and East Jerusalem was approximately 3.2 million and by 2050 it is projected to grow to 12 million. Use the exponential growth model , in which t is the number of years after 2000, to find the exponential growth function that models the data. (b) In which year will the Palestinian population be 9 million? SOLUTION (a) We use the exponential growth model in which t is the number of years after 2000. This means that 2000 corresponds to t = 0. At that time the Palestinian population was 3.2 million, Blitzer, Intermediate Algebra, 5e – Slide #5 Section 9.6

Exponential Growth so we substitute 3.2 for in the growth model: CONTINUED so we substitute 3.2 for in the growth model: We are given that the population will be 12 million in 2050. Because 2050 is 50 years after 2050, when t = 50 the value of A is 12. Substituting these numbers into the growth model will enable us to find k, the growth rate. We know that k > 0 because the problem involves growth. Use the growth model with When t = 50, A = 12. Substitute these numbers into the model. Blitzer, Intermediate Algebra, 5e – Slide #6 Section 9.6

Exponential Growth CONTINUED Isolate the exponential factor by dividing both sides by 3.2. Take the natural logarithm on both sides. Simplify the left side using Divide both sides by 50 and solve for k. The value of k, approximately 0.026, indicates a growth rate of about 2.6%. We substitute 0.026 for k in the growth model, , to obtain the exponential growth function for the Blitzer, Intermediate Algebra, 5e – Slide #7 Section 9.6

Exponential Growth CONTINUED Palestinian population. It is , where t is measured in years after 2000. (b) To find the year in which the Palestinian population will reach 9 million, substitute 9 for A in the model from part (a) and solve for t. This is the model from part (a). Replace A with 9. Divide both sides by 3.2. Take the natural logarithm on both sides. Blitzer, Intermediate Algebra, 5e – Slide #8 Section 9.6

Exponential Growth Simplify the left side using CONTINUED Simplify the left side using Divide both sides by 0.026 and solve for t. Because t represents the number of years after 2000, the model indicates that the Palestinian population in the West Bank, the Gaza Strip and East Jerusalem will reach 9 million by 2000 + 40, or in the year 2040. Blitzer, Intermediate Algebra, 5e – Slide #9 Section 9.6

Exponential Decay EXAMPLE The half-life of the radioactive element plutonium-239 is 25,000 years. If 16 grams of plutonium-239 are initially present, how many grams are present after 50,000 years? SOLUTION Since the half-life of plutonium-239 is 25,000 years, half of the 16 grams will still be present after 25,000 years. That is, 8 grams. After another 25,000 years, half of the 8 grams will still be present. That is, 4 grams. Since these two periods add up to the desired 50,000 years, the answer is 4 grams of plutonium-239. Now we will verify this answer using the exponential model, Blitzer, Intermediate Algebra, 5e – Slide #10 Section 9.6

Exponential Decay CONTINUED We begin with the decay model We know that k < 0 because the problem involves the decay of plutonium-239. After 25,000 years (t = 25,000), the amount of plutonium-239 present, A, is half the original amount, Thus, we can substitute for A in the exponential decay model. Begin with the exponential decay model. After 25,000 years (t = 25,000), Divide both sides by Blitzer, Intermediate Algebra, 5e – Slide #11 Section 9.6

Exponential Decay Take the natural log on both sides. CONTINUED Take the natural log on both sides. Simplify the right side using Divide both side by 25,000 and solve for k. Substituting for k in the decay model, , the model for plutonium-239 is Blitzer, Intermediate Algebra, 5e – Slide #12 Section 9.6

Exponential Decay CONTINUED We will replace t with 50,000. We will replace with 16 since the initial amount of plutonium-239 is 16 grams. This is the decay model for plutonium-239. Substitute t = 50,000 and Divide both sides by 16. Take the natural logarithm on both sides. Blitzer, Intermediate Algebra, 5e – Slide #13 Section 9.6

Exponential Decay Simplify the right side using Use the quotient rule. CONTINUED Simplify the right side using Use the quotient rule. Simplify. Add. Write in exponential form. Evaluate the left side. Therefore, there will be approximately 3.995 grams of plutonium-239 remaining after 50,000 years. Blitzer, Intermediate Algebra, 5e – Slide #14 Section 9.6

Exponential & Logarithmic Modeling y y x x y y x x Blitzer, Intermediate Algebra, 5e – Slide #15 Section 9.6

Percent of Miscarriages, by Age Percent of Miscarriages, by Age Exponential & Logarithmic Modeling EXAMPLE The following table presents a set of data. Use the data to do the following: Create a scatter plot for the data. Use the scatter plot to determine whether an exponential function, a logarithmic function, or a linear function is the best choice for modeling the data. Percent of Miscarriages, by Age Woman’s Age % of Miscarriages 22 9% 27 10% 32 13% Percent of Miscarriages, by Age Woman’s Age % of Miscarriages 37 20% 42 38% 47 52% Blitzer, Intermediate Algebra, 5e – Slide #16 Section 9.6

Exponential & Logarithmic Modeling CONTINUED SOLUTION (a) The scatter plot is given below. (b) Because the data in the scatter plot appear to increase more and more rapidly, the shape suggests that an exponential model might be a good choice. Blitzer, Intermediate Algebra, 5e – Slide #17 Section 9.6

Expressing an Exponential Model in Base e Rewriting Exponential Equations Expressing an Exponential Model in Base e Blitzer, Intermediate Algebra, 5e – Slide #18 Section 9.6

Rewriting Exponential Equations EXAMPLE Rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm and then round to three decimal places. SOLUTION Using the fact that , Therefore, an equivalent equation in terms of base e is Blitzer, Intermediate Algebra, 5e – Slide #19 Section 9.6

In summary… Exponential growth and decay models are given by the equation in which t represents time, is the initial amount present, that is, the amount present at time = 0, and A is the amount present at time t. If k < 0, the model describes decay and k is the decay rate. When using a graphing utility to model data, the closer the correlation coefficient is to -1 or 1, the better the model fits the data. Blitzer, Intermediate Algebra, 5e – Slide #20 Section 9.6