1 grain

800 billion tons

2t = Nt

One grain is to be doubled 64 times

Paper thickness 0.1 mm

Folds (10) 2^10

Number of papers thick. 1,024

x .1 mm

102.4 mm

How many inches is that?

102.4 mm x 1 cm x 1 inch 4.03 inches 10 mm 2.54 cm

How many miles thick are 42 folds?

272, 678 miles

How thick will the paper be after…

POPULATION

TIME

J - Curve

Unchecked biotic potential creates a “J- Curve” Environmental resistance forces the population down

S-Curve

When Environmental Resistance and Biotic Potential reach an equilibrium a carrying capacity is said to have been created.

Carrying capacity

Environmental Resistance

Population of reindeer on St. Matthew’s Island

What causes a population to grow exponentially?

Population Dynamics

e.g., population growth of the ring-necked pheasant

– 8 individuals introduced to Protection Island, Washington, in1937, increased to 1,325 adults in 5 years:• 166-fold increase!

Nt = number after specific timeNo = original numbere = r = ratet= time

Nt = No * er*t

Nt = 18.6No = 15.8 e = r = .023t= 7 years

EXAMPLE: A certain breed of rabbit was introduced onto a small island about 8 years ago. The current rabbit population on the island is estimated to be 4100, with a relative growth rate of 55% per year.(a) What was the initial size of the rabbit population?(b) Estimate the population 12 years from now.

4,100 = No * e (.55 * 8 )

4,10o = No

e (.55 * 8 )4,10o = 50 81.45)

Nt = 50 * e (.55

* 20 )

2,993,707

EXAMPLE: The population of the world in 2000 was 6.1 billion, and the estimated relative growth rate was 1.4% per year. If the population continues to grow at this rate, when will it reach 122 billion?

1.22 * 1011 = 6.1 * 109* e (.014 * t )

1.22 * 1011 = e (0.14 * t )

6.1 * 109

Ln(20)= e (0.14 * t )

3.0= 0.14 * t 3.0 = t 0.14

213.0 = t

EXAMPLE: A culture starts with 10,000 bacteria, and the number doubles every 40 min.(a) Find a function that models the number of bacteria at time t.(b) Find the number of bacteria after one hour.(c) After how many minutes will there be 50,000 bacteria?10, 000*e r(40) = 20, 000

e r(40) = 2

ln e r(40) = ln 2

40 r = .693r = .693 /40

r ≈ 0.01733

10, 000*e (0.0173 * 60) = Nt 10, 000*e (0.0173 * 60) = 28,235

(b)

(c)10, 000*e (0.0173*t) = 50, 000

e (0.0173*t) = 5

e (0.0173*t) = ln 5(0.0173*t) = 1.6

t = 1.6/0.0173 = 93

Logistic Growth ModelK = carrying capacity

K-N = a variable that influences population size and growth K

Carrying capacity = 600 N = 50

600 – 50 = .92 higher #, more resources are available 600

N = N *r K-N

t K

9.2 = 50 * .2 (.92) 30.0 = 300* .2 (.5) 9.1= 550 * .2 (.08)