Question

The US population officially reached 300 million on October 17, 2006 and was gaining 1 person each 11 seconds. If f (t) is the US population in millions t years after October 17, 2006, find f (0) and f' 0. Round your answer for f' (0) to three decimal places.

Answer 1

The US population function f(t) at time t = 0 (October 17, 2006) is 300 million, and the derivative f'(0), which represents the population growth rate at that moment, is approximately 2.867 million people per year.

Step-by-step explanation:

Using the given data that the US population officially reached 300 million on October 17, 2006, and was increasing by 1 person every 11 seconds, we can define the function f(t) to find the population in millions t years after this date. The value of f(0) represents the population at the time mentioned, which is 300 million. To find f'(0), we need to convert the rate of population increase into the change in population per year.

Since there are 31,536,000 seconds in a year, and the population increases by 1 person every 11 seconds, we can calculate the yearly growth as 31,536,000 seconds / 11 seconds/person ≈ 2,866,909 people per year. Converting this to millions gives us approximately 2.867 million people per year. Therefore, f'(0) is 2.867 million. This is the instantaneous rate of change of the population with respect to time at t = 0.