Respuesta :
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:
[tex]P(t) = (1436.53).(1.01395)^t[/tex]
where t = 0 corresponds to the year 1900 and P(t) is measured in millions.
Rate of increase of world population =
[tex]\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[/tex]
a) 1920
[tex]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[/tex]
Rate of increase = 26.66 millions per year
b) 1955
[tex]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[/tex]
Rate of increase = 42.64 millions per year
c) 2000
[tex]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[/tex]
Rate of increase = 79.53 millions per year
The rate of some function tells the increment in output values per unit increment in input.
The rate of increase of world population in 1920 was 17.02
In 1955 it was 42.64, and in 2000 it was 79.53 approx.
How to find the rate of an exponential function?
Suppose that the exponential function is given as
[tex]f(x) = a\times b^x[/tex]
Then its rate (first derivative) (assuming differentiable) with respect to x is given as
[tex]f'(x) = ab^x\ln(b)[/tex]
Since the given function for population measure as a function of time is
[tex]P(t) = (1436.53).(1.01395)^t[/tex]
Its rate is given as
[tex]P'(t) = (1436.53).(1.01395)^t\ln(1.01395) \approx 19.9 \times (1.01395)^t[/tex]
Since time was starting from t = 0 (year 1900), so at 1920, the value of t is 20.
Thus, rate of increase in world's population at year 1920 was
[tex]P'(t) =19.9 \times (1.01395)^t\\P'(20) = 19.9 \times (1.01395)^{20}\\\\P'(20) \approx 26.25[/tex]
Similarly,
[tex]P'(55) \approx 42.64 \\P'(100) \approx 79.53[/tex]
Thus,
The rate of increase of world population in 1920 was 17.02
In 1955 it was 42.64, and in 2000 it was 79.53 approx.
Learn more about rate of change of functions here;
https://brainly.com/question/15125607
