2. 1 grain 800 billion tons 2 t = N t One grain is to be doubled 64 times

3. 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…

4. 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

5. 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, in 1937, increased to 1,325 adults in 5 years: 166-fold increase!

6. N t = number after specific time N o = original number e = r = rate t= time N t = N o * e r*t N t = 18.6 N o = 15.8 e = r =.023 t= 7 years

7. 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 = N o * e (.55 * 8 ) 4,10o = N o e (.55 * 8 ) 4,10o = 50 81.45 ) N t = 50 * e (.55 * 20 ) 2,993,707

8. 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 * 10 11 = 6.1 * 10 9 * e (.014 * t ) 1.22 * 10 11 = e ( 0.14 * t ) 6.1 * 10 9 Ln(20)= e ( 0.14 * t ) 3.0= 0.14 * t 3.0 = t 0.14 213.0 = t

9. 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 =.693 r =.693 /40 r ≈ 0.01733 10, 000*e ( 0.0173 * 60) = N t 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

11. Logistic Growth Model K = 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)