Pre-Calculus

Current Populations

World United States 301 ,139 ,9471

Annual Population Growth Rate

World 1 .17 2

United States 0 .894 2 An exponential growth equation is given by the following formula

X X0ekt (1

Where X is the value of the population after a certain amount of time t , X0 is the initial value of the population , and k is the growth rate in decimal notation . Depending on the needed values , the time could be measured differently . For human populations , however , it is most commonly measured in years

p Based on current growth rates , how long will it take world population to reach exactly 7 billion ? How long to reach exactly 10 billion

In to solve the problem , an exponential equation governing population growth must first be set-up . In this case , we must assume that the annual population growth rate is static , that is , it does not vary through the years

Let X be the world population at a certain time t (in years , counting from current year , 2007 . Let Xo be the world population at the current year , which is equal to 6 ,605 ,046 ,992 . The value of k is equal to 1 .17 or 0 .0117 in decimal notation . Using (1 , equation governing the growth rate of the world population is

X 6 ,605 ,046 ,992e0 .0117t (2

Determine how long before the world population will reach 7 billion people

Given the population , we want to determine the amount of time when the population will reach the given value . X is 7 ,000 ,000 ,000 . Using (2

7 ,000 ,000 ,000 6 ,605 ,046 ,992e0 .0117t

Divide both sides by 6 ,605 ,046 ,992 in to isolate the exponent

1 .060 e0 .0117t

To solve for t , take the natural logarithm of both sides

ln (1 .060 ln (e0 .0117t

0 .058 0 .0117t

4 .956 years after 2007 , the population will be exactly 7 billion people This translates to roughly the year 2011 , in the month of November

Determine how long before the world population will reach 10 billion people

Using the same procedure as above , set up the exponential growth equation first

10 ,000 ,000 ,000 6 ,605 ,046 ,992e0 .0117t

1 .514 e0 .0117t

ln (1 .514 ln e0 .0117t

0 .415 0 .0117t

35 .47 years after 2007 , the world population will be exactly 10billion This corresponds to the year 2042 , in the month of May

Based on current growth rates , how long will it take for the U .S population to reach exactly 300 million ? How long for exactly 400 million

The exponential growth equation for this problem is given by

X 301 ,139 ,947e0 .00894t

Determine how long before the US population will reach 300 million people

Based on the given statistics , the US population has surpassed the 300 million mark in 2007 . Hence , the year when it passed the 300 million mark will have to be before 2007 . Solve for t in the same manner as the problems above...