Question

In the year 2000, the population of a certain country was 276 million with an estimated growth rate of 0.5% per year a. Based on these figures, find the doubling time and project the population in 2120 b Suppose the actual growth rates are ust 0.2 percentage points lower and higher than 0.5% per year 0.3% and 0.7%). What are the resulting doubling times and projected 2120 population? a. Let y(t) be the population of the country, in millions, t years after the year 2000. Give the exponential growth function for this country's population ytt) 276 e0.0050t (Use integers or decimals for any numbers in the expression. Round to four decimal places as needed.) What is the doubling time? 139 years (Round to the nearest whole number as needed.) What is the projected population for the year 2120? 502 milion people (Round to the nearest whole number as needed.) b. What is the doubling time when the growth rate is 0.3% per year? 231 years (Round to the nearest whole number as needed.) What is the projected population for the year 2120 when the growth rate is 0.3% per year? 395 million people (Round to the nearest whole number as needed.)

Answer 1

a. y(t) = 276 e^0.005t

we need to find t such that present population is doubled.

so, 2*276 = 276 e^0.005t

cancelling 276 from both sides, we get,

2 = e^0.005t

Taking natural log both sides,

ln2 = 0.005t

=> t = ln2/0.005 = 138.6294 = 139 (approx.)

(Note: t = ln2/r is doubling time formula, where r is rate of population growth)

Projected population in 2120 is 120 years ahead of the year 2000.

So, y(120) = 276 e^0.005*120 = 502.9047 = 503 (approx.)

b. According to a, we have  t = ln2/r (doubling time formula)

So, when r = 0.3% or 0.003

then doubling time t = ln2/0.003 = 231.0490 = 213 (approx.)

Projected population after 120 years from year 2000 when r = 0.003

y(120) = 276 e^0.003*120 = 395.5989 = 396 (approx.)