The population (in millions) of a certain country can be approximated by the function:
[tex]$P(x)=50 \cdot 1.02^x$[/tex]
where [tex]$x$[/tex] is the number of years after 2000. Which of the following calculations will tell in what year the population can be expected to reach 100 million?
A. [tex]$\frac{\ln (2)}{\ln (1.02)}+2000$[/tex]
B. [tex]$\frac{\ln (2)}{\ln (1.02)}$[/tex]
C. [tex]$\ln \left(\frac{2}{1.02}\right)$[/tex]
D. [tex]$\ln \left(\frac{2}{1.02}\right)+2000$[/tex]
Answer (2)
Set the population function equal to 100 million: 100 = 50 ⋅ 1.0 2 x .
Divide both sides by 50: 2 = 1.0 2 x .
Take the natural logarithm of both sides: ln ( 2 ) = x ln ( 1.02 ) .
Solve for x and add to 2000 to find the year: 2000 + l n ( 1.02 ) l n ( 2 ) . The answer is ln ( 1.02 ) ln ( 2 ) + 2000 .
Explanation
Understanding the Problem We are given the population function P ( x ) = 50 ⋅ 1.0 2 x , where P ( x ) is the population in millions and x is the number of years after 2000. We want to find the year when the population reaches 100 million.
Setting up the Equation To find the year when the population reaches 100 million, we set P ( x ) = 100 and solve for x . This gives us the equation: 100 = 50 ⋅ 1.0 2 x
Simplifying the Equation Divide both sides of the equation by 50: 50 100 = 1.0 2 x 2 = 1.0 2 x
Applying Logarithms Take the natural logarithm of both sides of the equation: ln ( 2 ) = ln ( 1.0 2 x )
Using the Power Rule of Logarithms Use the power rule of logarithms, which states that ln ( a b ) = b ln ( a ) :
ln ( 2 ) = x ln ( 1.02 )
Solving for x Solve for x by dividing both sides by ln ( 1.02 ) :
x = ln ( 1.02 ) ln ( 2 )
Finding the Year Since x is the number of years after 2000, the year when the population reaches 100 million is 2000 + x . Therefore, the year is: 2000 + ln ( 1.02 ) ln ( 2 )
Final Calculation The calculation that will tell us in what year the population can be expected to reach 100 million is: ln ( 1.02 ) ln ( 2 ) + 2000
Examples
Population growth models are used in many real-world applications, such as urban planning, resource management, and public health. For example, city planners use population projections to estimate the demand for housing, transportation, and other services. Similarly, public health officials use population growth models to predict the spread of infectious diseases and to plan vaccination campaigns. Understanding exponential growth, as demonstrated in this problem, is crucial for making informed decisions in these areas. The formula P ( x ) = 50 ⋅ 1.0 2 x is a simplified version of these models, but it captures the essential feature of exponential growth, where the population increases by a constant percentage each year. In our case, we found that the population is expected to reach 100 million in the year 2000 + l n ( 1.02 ) l n ( 2 ) ≈ 2035 .
To find the year when the population reaches 100 million, we solve the equation set from the population function. The calculation needed is l n ( 1.02 ) l n ( 2 ) + 2000 , which gives us the correct answer of Option A. This represents the number of years after 2000 plus the year 2000 itself.
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