Modeling with Exponential and Logarithmic Functions

The mathematical model for exponential growth or decay is given by f (t) = A 0 e kt or A = A 0 e kt. If k > 0, the function models the amount or size of a growing entity. A 0 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. If k < 0, the function models the amount or size of a decaying entity. A 0 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. The mathematical model for exponential growth or decay is given by f (t) = A 0 e kt or A = A 0 e kt. If k > 0, the function models the amount or size of a growing entity. A 0 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. If k < 0, the function models the amount or size of a decaying entity. A 0 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. decreasing A0A0 x y increasing y = A 0 e kt k > 0 x y y = A 0 e kt k < 0 A0A0 Exponential Growth and Decay Models

The graph below shows the growth of the Mexico City metropolitan area from 1970 through In 1970, the population of Mexico City was 9.4 million. By 1990, it had grown to 20.2 million. Find the exponential growth function that models the data. By what year will the population reach 40 million? Population (millions) Year Example

Solution a. We use the exponential growth model A = A 0 e kt in which t is the number of years since This means that 1970 corresponds to t = 0. At that time there were 9.4 million inhabitants, so we substitute 9.4 for A 0 in the growth model. A = 9.4 e kt We are given that there were 20.2 million inhabitants in Because 1990 is 20 years after 1970, when t = 20 the value of A is 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. A = 9.4 e kt Use the growth model with A 0 = = 9.4 e k20 When t = 20, A = Substitute these values. Example cont.

Solution We substitute for k in the growth model to obtain the exponential growth function for Mexico City. It is A = 9.4 e 0.038t where t is measured in years since Example cont. 20.2/ 9.4 = e k20 Isolate the exponential factor by dividing both sides by 9.4. ln(20.2/ 9.4) = lne k20 Take the natural logarithm on both sides = k Divide both sides by 20 and solve for k. 20.2/ 9.4 = 20k Simplify the right side by using ln e x = x.

Solution b. To find the year in which the population will grow to 40 million, we substitute 40 in for A in the model from part (a) and solve for t. Because 38 is the number of years after 1970, the model indicates that the population of Mexico City will reach 40 million by 2008 ( ). A = 9.4 e 0.038t This is the model from part (a). 40 = 9.4 e 0.038t Substitute 40 for A. Example cont. ln(40/9.4) = lne 0.038t Take the natural logarithm on both sides. ln(40/9.4)/0.038 =t Solve for t by dividing both sides by ln(40/9.4) =0.038t Simplify the right side by using ln e x = x. 40/9.4 = e 0.038t Divide both sides by 9.4.

Use the fact that after 5715 years a given amount of carbon-14 will have decayed to half the original amount to find the exponential decay model for carbon-14. In 1947, earthenware jars containing what are known as the Dead Sea Scrolls were found by an Arab Bedouin herdsman. Analysis indicated that the scroll wrappings contained 76% of their original carbon-14. Estimate the age of the Dead Sea Scrolls. Solution We begin with the exponential decay model A = A 0 e kt. We know that k < 0 because the problem involves the decay of carbon-14. After 5715 years (t = 5715), the amount of carbon-14 present, A, is half of the original amount A 0. Thus we can substitute A 0 /2 for A in the exponential decay model. This will enable us to find k, the decay rate. Text Example

Substituting for k in the decay model, the model for carbon-14 is A = A 0 e – t. k = ln(1/2)/5715= Solve for k. 1/2= e k5715 Divide both sides of the equation by A 0. Solution A 0 /2= A 0 e k5715 After 5715 years, A = A 0 /2 ln(1/2) = ln e k5715 Take the natural logarithm on both sides. ln(1/2) = 5715k ln e x = x. Text Example cont.

Solution The Dead Sea Scrolls are approximately 2268 years old plus the number of years between 1947 and the current year. A = A 0 e t This is the decay model for carbon A 0 = A 0 e t A =.76A 0 since 76% of the initial amount remains = e t Divide both sides of the equation by A 0. ln 0.76 = ln e t Take the natural logarithm on both sides. ln 0.76 = t ln e x = x. Text Example cont. t=ln(0.76)/( ) Solver for t.

Logistic Growth Model The mathematical model for limited logistic growth is given by Where a, b, and c are constants, with c > 0 and b > 0. c is the limiting size!

a)How many people became ill when the epidemic began? This function describes the number of people who have become ill with influenza t weeks after its initial outbreak in a town with 30,000 inhabitants

b)How many people were ill be the end of the 4 th week?

This function describes the number of people who have become ill with influenza t weeks after its initial outbreak in a town with 30,000 inhabitants c)What is the limiting size of f(t)? c is the limiting size of a logistic growth model; 30,000

Newton’s Law of Cooling The temperature, T, of a heated object at time t is given by T = C + (T 0 - C)e kt Where C is the constant temperature of the surrounding medium, T 0 is the initial temperature of the heated object, and k is a negative constant that is associated with the cooling object.

A cake removed from the oven has a temp. of 210 degrees. It is left to cool in a room that has a temp of 70 degrees. After 30 minutes, the temp of the cake is 140 degrees. a)Find a model for the temp, T, after t minutes. b)What is the temp of the cake after 40 minutes? c)When will the temp be 90 degrees?

A cake removed from the oven has a temp. of 210 degrees. It is left to cool in a room that has a temp of 70 degrees. After 30 minutes, the temp of the cake is 140 degrees. a)Find a model for the temp, T, after t minutes.

A cake removed from the oven has a temp. of 210 degrees. It is left to cool in a room that has a temp of 70 degrees. After 30 minutes, the temp of the cake is 140 degrees. b)What is the temp of the cake after 40 minutes?

A cake removed from the oven has a temp. of 210 degrees. It is left to cool in a room that has a temp of 70 degrees. After 30 minutes, the temp of the cake is 140 degrees. c)When will the temp be 90 degrees?

When a murder is committed, the body, originally 37C, cools according to Newton’s Law of Cooling. Suppose on the day of a very difficult exam your math teacher’s body is found at 2pm with a body temperature of 32C in a room with a constant temperature of 20C. 3 hours later the body temperature is 27C. Being the bright student you are, find an equation to model this situation and determine at what time the murder was committed.

Expressing an Exponential Model in Base e. y = ab x is equivalent to y = ae (lnb)x

Example The value of houses in your neighborhood follows a pattern of exponential growth. In the year 2000, you purchased a house in this neighborhood. The value of your house, in thousands of dollars, t years after 2000 is given by the exponential growth model V = 125e.07t When will your house be worth $200,000?

Example Solution: V = 125e.07t 200 = 125e.07t 1.6 = e.07t ln1.6 = ln e.07t ln 1.6 =.07t ln 1.6 /.07 = t 6.71 = t