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