Exponential Growth

Rabbits are famous for rapid reproduction.

Let’s build a model of population growth.

To begin with, let’s count only the female rabbits.

Let’s assume they reproduce at specified times, so we can keep track of the generations.

Assume each female of reproductive age produces one additional female each generation.

Assume females mature by the second generation after they are born.

Ignore deaths.

1

1

3

2

5

8

13

Fibonacci numbers

LogarithmsJohn Napier, Baron of Merchistoun (1550 –1617), asked how to write a number x as a power of 10:

x = 10?

x = 10log(x)

e.g., 100 = 102

log(100) = 2 100,000 = 105

log(100,000) = 50.001 = 10-3

log(0.001) = -3

1 = 100

log(1) = 0

ar as = ar+s

42 43 = 4x4x4x4x4 = 42+3= (4x4)x(4x4x4)

x = 10log(x)

x y = 10log(x) 10log(y) = 10log(x)+log(y)

But: x y = 10log(xy)

Comparing exponents, we find:

log(xy) = log(x) + log(y)

log(x)+log(y)

log(xy)

It is possible to use different numbers instead of 10.

The most common is e ≈ 2.71828….

x = eln(x)

ln(xy) = ln(x) + ln(y)

x = eln(x)

ln(er) = r

Two other useful rules:

(ar)s = ars a0 = 1

Why logarithms?Ease of calculation

Actually occur in nature

Logarithmic spirals

Neuron response (the logarithm of the intensity of the stimulus)

Decibels

The Richter scale

Bacteria reproduce by dividing.

1 Generation 02 Generation 14 Generation 28 Generation 3

Generation n2n

: : : :

Number of bacteria after n generations is

2n = (eln(2))n

= eln(2)n

Note that n is essentially the elapsed time.

We write a “general exponential function” as f(t) = Cekt.

f(t) = Cekt

Consider f(0) = Ce0 = C x 1 = C,

The constant k is called the “growth constant”.

so C is the “initial value”: C = f(0).

Consider a culture of bacteria which weighs 3 mg.

Suppose that 3 hours later it weighs 7 mg.

How much will it weigh five hours after that?

Solution: Assume the growth is exponential:

P(t) = Cekt.

We know the initial value: C = 3. So P(t) = 3ekt.

We also know that P(3) = 7, i.e., 3ek3 = 7.

So e3k = 7/3. Taking ln’s, we find 3k = ln(7/3),so k = ln(7/3)/3, and P(t) = 3eln(7/3)/3 t.

In particular, P(8) = 3eln(7/3)/3 x 8 ≈ 28.7 mg.

Suppose we start with 5 g of yogurt (1 tsp), and suppose the amount doubles every 45 minutes.

So Y(t) = 5ekt.

Also, Y(0.75) = 10, i.e., 5e0.75 k = 10,

so e0.75 k = 2, and, taking ln’s, we find

0.75 k = ln(2), and Y(t) = 5eln(2)/0.75 t.

After 6 hours, we have Y(6) = 5eln(2)/0.75 6

= 1280 g,

a little over a litre.

What happens after 72 hours?

Y(72) = 5eln(2)/0.75 x 72 ≈ 4 x 1029 g

= 4 x 1026 kg

= 4 x 1023 tonnes

= 4 x 1023 m3 (cubic metres)

4 x 1023 m3 (cubic metres)

Length of each side is

cube root of 4 x 1023

= (4 x 1023)1/3

≈ 74,000,000 m

= 74,000 km

74,000 km

Diameter of earth?

12,000 km

graph of y = ex ?

5 10 15 20 25

graph of y = ex ?

5 10 15 20 25

e25 ≈ 7.2 x 1010

7.2 x 1010 cm

= 7.2 x 108 m

= 7.2 x 105 km = 720,000 km

How far is the moon?

Chess was invented somewhere in what is now Persia or India, but nobody knows exactly where. Written records go back at least fourteen centuries, and to about 900 AD in Europe.

It must have come from a culture with horses.

The phrase “check mate” is a corruption of the old Persian for “the king is dead”.

There are also Chinese precursors of chess.

Legend has it that an ancient king was so delighted with the invention of chess that he offered its inventor whatever he asked.

The inventor of chess thought for a moment, and then smiled.

“I don’t want much,” he explained. “Take a chessboard. Just put one grain of wheat on the first square, two on the second, four on the third, and so on, doubling each time. That will be my reward.”

It works out to 264-1 grains of wheat, which is more than the modern annual global harvest.

Compound InterestDeposit an amount of “principal” in a bank.

“Simple interest” refers to a payment by the bank of a fixed percentage of the principal when the loan is returned.

“Compound interest” refers to payment of interest in stages, with later interest payments including interest on the earlier interest payments.

For example, if you invest $10,000 at 5% simple interest for a year, at the end of the year, you get $10,500.

But if the interest is compounded every six months, you get 2.5% every half year: $10,250.00 after six months, and then at the end of the year, 1.025 x $10,250.00 = $10,506.25.

For simple interest, if the interest rate is r%, the amount is multiplied by

1 +

If the interest rate is r% and the interest is compounded n times, then each time interest is paid, the amount is multiplied by

1 +

r

100

r/n

100

By the end of the term of the loan, the original amount has been multiplied by

( 1 + )r/n

100

n

By the end of the term of the loan, the original amount has been multiplied by

( 1 + )r/n

100

n

The larger n is, i.e., the more often the interest is compounded, the larger the total amount repaid.

It is possible to compound quarterly, monthly, daily, hourly, or even more often.

But as n gets larger and larger, the factor above does not get arbitrarily large. It tends to a “limit”, which is er/100.

This is known as “continuous compounding”, and the amount of money in the account increases according to exponential growth.

Nuclear PhysicsAn atom consists of a heavy nucleus surrounded by lighter electrons.

The nucleus is made up of positive protons and electrically neutral neutrons. The electrons are negative, and the charge of each electron is exactly equal and opposite to that of each proton. The electrons and protons attract one another.

Chemistry involves one or more atoms “sharing” their electrons.

Sometimes, it is possible to have two atoms with the same numbers of protons and electrons, but different numbers of neutrons.

Because their electric properties are the same, and specifically because the number of electrons is the same, they have the same chemical behaviour.

Two such atoms are called different “isotopes” of the same element.

An important example is carbon. The common isotope is C12, which has 6 protons and 6 neutrons.

But there is another isotope, C14, which has 6 protons and 8 neutrons.

Because C12 and C14 have the same number of protons and electrons, they behave the same way in chemical reactions.

But C14 is radioactive. Its nucleus is not stable, and from time to time, one of the nuclei in a sample of C14 decays, disintegrating.

The rate at which C14 decays is very predictable. Half of the nuclei in a sample will decay in 5730 years (the “half-life” of C14 ).

In the atmosphere, C14 is created when nitrogen is bombarded with radiation from space.

The result is that the carbon dioxide in the atmosphere contains a small proportion of C14.

Plants ingest CO2 and metabolize it. Since the processes involved are chemical in nature, they have exactly the same effect on the two isotopes.

Consequently, the carbon out of which plants are made contains the same small proportion of C14 .

When animals eat plants and digest them, the process is again chemical, so the carbon in animals contains the same proportion of C14 .

When an organism dies, it stops ingesting new carbon.

The C14 begins its slow radioactive decay, so gradually the proportion of C14 decreases.

By measuring the proportion of C14 remaining, scientists can calculate the time that has elapsed since the organism died.

This is the basis of carbon dating.

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Atmospheric nuclear tests have disrupted the amount of C14 in the atmosphere, so this technique is no longer available for organisms which were alive since the nuclear age began.