The given function models growth of the population: population t is

P(t)=L/1+(L-Po/Po) e -kt

where Po is population at time t=0, k is growth rate, and L is limiting value, defined as maximum possible population which environment can support. Functions which look like this always have the moment when change in growth starts to reduce, called moment of diminishing returns. It so occurs that moment of diminishing returns of P9t) as given above, occurs when P(t)=L/2

Human population on Earth is presently about 7.094 billion. Earth's human population in year 1800 was about 1 billion. Scientist say that liberal estimate of how many people possibly survive on planet at one time is about 10 billion.

(a). Suppose human population on Earth is modeled by function above, determine function H(t) which outputs how many billions of humans will be on Earth at year t, using 10 billions limiting value. Find k to 6 decimal places. Let t=0 correspond to year 1800.

(b) According to the function from part (a) find human population in year 2020?

(c) In what year did point of diminishing returns happen?

(d) There are also few who say that once human population of Earth reaches 8 billion, sociological conditions will be such that population will no longer modeled by function: but instead will start to drop, marking beginning of apocalypse. Assume this theory is true, in what year would apocalypse occur?