Hence, Maximum Population Will Become in 2080

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Assume the world population will continue to grow exponentially with a growth constant k=0.0132 (corresponding to a doubling time of about 52 years), it takes 12 acre of land to supply food for one person, and there are 13,500,000 square miles of arable land in in the world. How long will it be before the world reaches the maximum population? Note: There were 6.06 billion people in the year 2000 and 1 square mile is 640 acres.

Answer: The maximum population will be reached some time in the year . Hint: Convert .5 acres of land per person (for food) to the number of square miles needed per person. Use this and the number of arable square miles to get the maximum number of people which could exist on Earth. Proceed as you have in previous problems involving exponential growth.

Maximum population = 13500000*640/0.5

Hence, 13500000*640/0.5 =6.06*109exp(0.0132t) or t = 79.38.

Hence, maximum population will become in 2080.

Some time in the future a human colony is started on Mars. The colony begins with 50000 people and grows exponentially to 150000 in 150 years. Give a formula for the size of the human population on Mars as a function of t= time (in years) since the founding of the original colony

P = Poexp(kt)

P=150000, t = 150 years. Hence, 150000=50000exp(150k)

Or, exp(150k) = 150000/50000=3

Or, k = ln3/150 = 7.324*10-3

Hence, P = 50000exp(7.324*10-3t) where t is in years.

Assuming the population continues to grow exponentially, how long will it take to reach a size of 750000? 75000=50000exp(7.324*10-3t)

Or, exp (7.324*10-3t)=75000/50000=1.5

Or, t = ln1.5/7.324*10-3=55.361 years.

What is the rate of change of the size of the population 350 years after the founding of the original colony? We have dP/dt = 50000*7.324*10-3*exp(7.324*10-3t)

-3 -3 Hence, dP/dt|350 = 50000*7.324*10 *exp(7.324*10 *350)=4753.23 persons/year.

The total number of people infected with a virus often grows like a logistic curve. Suppose that 25 people originally have the virus, and that in the early stages of the virus (with time, t, measured in weeks), the number of people infected is increasing exponentially with k=1.6. It is estimated that, in the long run, approximately 6750 people become infected. (a) Use this information to find a logistic function to model this situation.

Logistic curve is given by P= A/[1+Bexp(-1.6t)].

Here P = 6750 at t = ∞. Hence, 6750 = A.

P = 25 at t = 0; hence, 25 = 6750/[1+B) or, B = (6750/25)-1 = 269.

Hence, P = 6750/[1+269exp(1.6t)] is the required equation.

(b) Sketch a graph of your answer to part (a). Use your graph to estimate the length of time until the rate at which people are becoming infected starts to decrease. What is the vertical coordinate at this point? vertical coordinate =

Plot is given below.

Vertical ordinate = 3400