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Modeling with Exponential Functions

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y

y

increasing

y = A0ekt

k > 0

y = A0ekt

k < 0

A0

x

x

- The mathematical model for exponential growth or decay is given by
- f (t) = A0ekt or A = A0ekt.
- If k > 0, the function models the amount or size of a growing entity. A0 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. A0 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.

Exponential Growth and Decay Models

decreasing

A0

30

25

20

15

Population (millions)

10

5

1970

1980

1990

2000

Year

The graph below shows the growth of the Mexico City metropolitan area from 1970 through 2000. In 1970, the population of Mexico City was 9.4 million. By 1990, it had grown to 20.2 million.

Example

- Find the exponential growth function that models the data.
- By what year will the population reach 40 million?

Solution

a. We use the exponential growth model

A = A0ekt

in which t is the number of years since 1970. This means that 1970 corresponds to t = 0. At that time there were 9.4 million inhabitants, so we substitute 9.4 for A0 in the growth model.

A = 9.4 ekt

Example cont.

We are given that there were 20.2 million inhabitants in 1990. Because 1990 is 20 years after 1970, when t = 20 the value of A is 20.2. 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 ektUse the growth model with A0 = 9.4.

20.2 = 9.4 ekâ€¢20When t = 20, A = 20.2. Substitute these values.

Solution

20.2/ 9.4 = ekâ€¢20Isolate the exponential factor by dividing both sides by 9.4.

Example cont.

ln(20.2/ 9.4) = lnekâ€¢20Take the natural logarithm on both sides.

20.2/ 9.4 = 20kSimplify the right side by using lnex = x.

0.038 = kDivide both sides by 20 and solve for k.

We substitute 0.038 for k in the growth model to obtain the exponential growth function for Mexico City. It is A = 9.4 e0.038t where t is measured in years since 1970.

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.

Example cont.

A = 9.4 e0.038t This is the model from part (a).

40 = 9.4 e0.038t Substitute 40 for A.

40/9.4 = e0.038t Divide both sides by 9.4.

ln(40/9.4) = lne0.038t Take the natural logarithm on both sides.

ln(40/9.4) =0.038t Simplify the right side by using lnex = x.

ln(40/9.4)/0.038 =t Solve for t by dividing both sides by 0.038

Because 38 is the number of years after 1970, the model indicates that the population of Mexico City will reach 40 million by 2008 (1970 + 38).

Isotopes

For any neutral atom, the number of electrons will equal the number

of protons. It is the number of protons that determines which element

we have. If an element has six protons, the element must be carbon.

How many neutrons does carbon have? Carbon usually has six

neutrons, but some atoms of carbon occasionally have seven or even eightneutrons! These three types of carbon are called C-12, C-13 and C-14

and are said to be isotopes of carbon. Isotopes of an atom have the

same number of protons, but different numbers of neutrons. The small

superscripts in the symbols refer to the mass number of the atom,

which allows us to calculate the number of neutrons. Because these

isotopes all have the same number of protons (or atomic number),

they are all forms of carbon, and behave like carbon in their chemical

reactions.

Carbon 14 Dating

Ancient remains containing carbon are sometimes dated using a technique called carbon dating which makes use of the C-14 isotope.

To understand how carbon -14 dating works, you first have to understand what carbon-14 is and what part it plays in our biosphere. All living creatures are made in part of carbon. As they live and grow and interact with their environments they consume more and more carbon. By far, the most abundant form (isotope) of carbon is carbon-12. Carbon-12 is a stable isotope; that is, it doesn't decay naturally. Carbon-14 on the other hand is an unstable isotope; that is, it decays naturally over time. Carbon-14 is also relatively rare.

As creatures consume carbon from their environment and incorporate it into their bodies, they consume both carbon-12 and carbon-14. When a creature dies, it ceases to consume more carbon. Since carbon-12 doesn't naturally decay while carbon-14does, once a creature stops incorporating more carbon into its body, the ratio of carbon-12 to carbon-14 in its body begins to change, with less carbon-14 per carbon-12 as time passes.

- In 1947, earthenware jars containing what are known as the Dead Sea Scrolls were found by an Arab Bedouin herdsman. Analysis using the technique of carbon dating indicated that the scroll wrappings contained 76% of their original carbon-14.
- 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.
- Estimate the age of the Dead Sea Scrolls in 2010

Activity

Solution

We begin with the exponential decay model

A = A0ekt

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 A0.

Thus we can substitute A0/2 for A in the exponential decay model. This will enable us to find k, the decay rate.

A0/2= A0ek5715After 5715 years, A = A0/2

Solution cont.

1/2= ekt5715Divide both sides of the equation by A0.

ln(1/2) = ln ek5715Take the natural logarithm on both sides.

ln(1/2) = 5715klnex = x.

k = ln(1/2)/5715=-0.000121Solve for k.

Substituting for k in the decay model, the model for carbon-14 is

A = A0eâ€“0.000121t.

A = A0e-0.000121tThis is the decay model for carbon-14.

Solution cont.

0.76A0 = A0e-0.000121tA = .76A0 since 76% of the initial amount remains.

0.76 = e-0.000121tDivide both sides of the equation by A0.

ln 0.76 = lne-0.000121tTake the natural logarithm on both sides.

ln 0.76 = -0.000121tlnex = x.

t=ln(0.76)/(-0.000121)Solver for t.

The Dead Sea Scrolls are approximately 2268 years old plus the number of years between 1947 and 2010. That is 2331 years old