Question

In this example we modeled the world population from 1900 to 2010 with the exponential function P(t) = (1436.53) · (1.01395)t where t = 0 corresponds to the year 1900 and P(t) is measured in millions. According to this model, what was the rate of increase of world population in 1920? 1955? 2000? (Round your answers to two decimal places.)

Answer 1

Answer:

a) Rate of increase = 26.66 millions per year

b) Rate of increase = 42.64 millions per year

c) Rate of increase = 79.53 millions per year

Step-by-step explanation:

We are given that population is modeled with the exponential function:

$P(t) = (1436.53).(1.01395)^t$

where t = 0 corresponds to the year 1900 and P(t) is measured in millions.

Rate of increase of world population =

$\displaystyle\frac{d(a^x)}{dx} = a^xln(a)\\\\P'(t) = (1436.53)(1.01395)^tln(1.01395)\\\\P'(t) = (1436.53)ln(1.01395)(1.01395)^t$

a) 1920

$P'(1920-1900) = (1436.53)(ln(1.01395))(1.01395)^{(1920-1900)}\\P'(20) = (1436.53)(ln(1.01395))(1.01395)^{(20)}\\P'(20) \approx 26.66$

Rate of increase = 26.66 millions per year

b) 1955

$P'(1955-1900)=(1436.53)(ln(1.01395))(1.01395)^{(1955-1900)}\\P'(55) = (1436.53)(ln(1.01395))(1.01395)^{(55)}\\P'(55) \approx 42.64$

Rate of increase = 42.64 millions per year

c) 2000

$P'(2000-1900) = (1436.53)(ln(1.01395))(1.01395)^{(2000-1900)}\\P'(100) = (1436.53)(ln(1.01395))(1.01395)^{(100)}\\P'(100) \approx 79.53$

Rate of increase = 79.53 millions per year