Question

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. Use this model to find the rate of change of world population (in million/yr) in 1920, 1954, and 2000. (Round your answers to two decimal places):

a) Rates for 1920, 1954, and 2000
b) Exponential function calculations
c) Population change over time
d) World population growth analysis

Answer 1

To calculate the rate of change of world population for given years using the exponential function P(t), we derive the function to get the growth rate dP/dt and evaluate it at the respective years: 1920, 1954, and 2000.

Step-by-step explanation:

To find the rate of change of world population in 1920, 1954, and 2000, using the exponential function
`P(t) = 1436.53 . (1.01395)^t`, we first need to calculate the derivative of P with respect to t, which gives us the rate of growth at any given time t.

The derivative dP/dt is given by the product of the initial population, the rate constant, and the exponential function itself. Thus, dP/dt = 1436.53 ·
`ln(1.01395) . (1.01395)^t`. We can then plug in the values of t that correspond to the years 1920, 1954, and 2000 to find the rates of growth for those years.

For t = 20 (year 1920):

`dP/dt = 1436.53 . ln(1.01395) . (1.01395)^(^2^0^)`,
which after calculation gives a certain value (in million/yr).

For t = 54 (year 1954):

`dP/dt = 1436.53 . ln(1.01395) . (1.01395)^(^5^4^)`,
which after calculation gives another value (in million/yr).

For t = 100 (year 2000):

`dP/dt = 1436.53 . ln(1.01395) .(1.01395)^(^1^0^0^)`,
which after calculation gives yet another value (in million/yr).

We can conclude that the population change over time is a result of the exponential function calculations and can be analyzed to understand world population growth.