Question

The population of the country of Oz was 600,000 in the year 2010. The population is expected to grow by a factor

of 5% annually. The annual food supply of Oz is currently sufficient for a population of 700,000 people and is
increasing at a rate that will supply food for an additional 10,000 people per year.
c. At what point does the population exceed the food supply? Justify your response.

Answer 1

After 5 years (2015)

Explanation:

The population is increasing 5% per year, so the first year it will be 1.05 of the initial, the next year 1.05x1.05 of the initial, and then successively, so, for n years:

P(n) = 600,000*(1.05)ⁿ

The food suply increase 10,000 people per year, so the first year will be 10,000 + the initial, the next year 10,000 + 10,000 + the initial, and then successively, so, for n years:

F(n) = 70,000 + 10,000*n

So

P(n) > F(n)

600,000*(1.05)ⁿ > 70,000 + 10,000*n

(1.05)ⁿ > 1.167 + 0.0167n

Solving in a computer:

n > 4.42

So, after 5 years the population exceed the food supply.