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continues, until the bottle is completely ‘ full ’ at 1 pm. We encourage teachers to ask the following questions because the questions help students un derstand the counter intuitive nature of the expo nential function, namely, powers of 2. Question: at what time was the bottle half full?

time for the current population of 7.5 billion peo ple was ≈ 70/1.1 ≈ 62.5years. In 2022, the world population was about 7.9 billion and in 2021 the growth rate dropped to 1.03%. This means, the doubling time is ≈ 68 years. We are now ready to explore exponential models for the consumption of finite resources. We are now in a position to discuss the problem that so vexed Dr. Bartlett so many years ago, the consumption of finite resources. Unlike the energy sources that come from solar, wind, hydrothermal or hydroelectric power, oil and coal are finite re sources. This means the concept of sustainable growth based on fossil fuels is not possible. We show this using various mathematical models. This section is suitable for students who know integral calculus for exponential functions. Let C 0 be the initial rate of consumption (i.e., at t = 0), and furthermore suppose that this rate doubles every decade due to improvements in extraction technology. If time is measured in decades, it fol lows that C ( t ) = C 0 (2 t ) = C 0 ( ) = C 0 ( ) = C 0 ( ) , where k = ln2 ≈ 0.69315. Given this, the amount of resource consumed in the interval [0, t ] is Mathematical Models for Consumption The basic model

Answer: 12:59 pm

Question: How full is the bottle at 12:55 pm?

Answer: (1/2) =1/32 and room for growth.

Plenty of time

Question: How full is the bottle at 12:50 pm?

Answer:

Seemingly almost empty for the last question, in reality the bacteria are living on borrowed time because their bottle universe will “ end ” in 10 minutes. Next, we examine a more general popula tion growth problem, in which we ask the question, how long will a given population take to double in size? For this example, students require knowledge us ing logarithmic functions and in particular, the nat ural logarithmic function. Suppose that an initial population N 0 , for example, a city or country grows simplistically according to the equation kt , where t is time, years, and k is the growth rate, such as 5%, represented as k =0.05. We can calculate the time for the population to double ( t d ) in size as shown below: A Population Activity N ( t ) = N 0 e

C ( t ) =

= C 0

= (

). (3)

Suppose that the size of the resource is R , then the time T e to consume it, the expiration of that re source, occurs when C ( T e ) = R . That is, when

) = R , or =

(

(4)

We will use this result with recent data. In 2020, about 480 million tons of coal were consumed in the United States, this is C 0 . The recoverable coal reserves remaining are currently estimated to be about 250 billion tons, this is R . Suppose that the U.S. was able to reduce its consumption of coal. Since many people believe coal reserves are unlim ited, we ask the question, what should be the re duction rate k per decade for these reserves to last forever? Note that, in this case k < 0, so we ask specifically, what choice of k will yield T e = ∞? k = - │ k │ and T e → ∞ → → ∞, i.e., →

yrs.

This means the doubling time to go from 2 to 4 is the same as the doubling time to go from 20 mil lion to 40 million. For an arbitrary growth rate of k %, the doubling time t d ≈ 70/ k . The world popula tion growth rate was 1.12% in 2017. If the growth is exponential, then the doubling t d ≈ 70/ k. The

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