Ch. 12 question

Question

In 1973, the total population of a species reached 44 billion. The total population of the species reached 88 billion in 1998. If the total population continues to grow exponentially at this​ rate, what will the population be in

2047​?

Answer 1

Let P(t) be the population in billions, as a function of time (in years)

Let P_o be the population in 1973 = 44

The question tells you that the population is growing exponentially. Therefore, the equation to calculate the population after t years is:

P(t) = P_oe^kt

In this equation, k is a rate constant. Let's solve for k with the information we have:

In 1998, the population was 88 billion. This is 25 years after 1973, when the initial population was 44 billion. Therefore, t = 25:

88 = 44e^k(25)

Divide both sides by 44

2 = e^{k(25 year)}

Take the natural logarithm (ln) of both sides

ln(2) = (25 year)k

k = (ln2)÷(25 year) = 0.0277 year^-1

For the year 2047, t = 74 years. (2047 - 1973 = 74)

Let's solve for P:

P = 44e^0.0277(74)

P ≈ 342 billion