The population (in millions) of a certain island can be approximated by the function [tex]$P(x)=50(1.05)^x$[/tex], where [tex]$x$[/tex] is the number of years since 2000. In which year will the population reach 200 million?
A. 2066
B. 2002
C. 2015
D. 2028
Answer (2)
Set up the equation 200 = 50 ( 1.05 ) x to model the population growth.
Simplify the equation to 4 = ( 1.05 ) x .
Solve for x using logarithms: x = l n ( 1.05 ) l n ( 4 ) ≈ 28.4134 .
Add x to 2000 to find the year: 2000 + 28.4134 = 2028.4134 , so the population reaches 200 million in the year 2028 .
Explanation
Understanding the Problem The problem states that the population of an island is modeled by the function P ( x ) = 50 ( 1.05 ) x , where x is the number of years since 2000. We need to find the year when the population reaches 200 million.
Setting up the Equation To find the year when the population reaches 200 million, we set P ( x ) = 200 and solve for x . This gives us the equation: 200 = 50 ( 1.05 ) x
Simplifying the Equation Divide both sides of the equation by 50: 50 200 = ( 1.05 ) x 4 = ( 1.05 ) x
Applying Logarithms Take the natural logarithm of both sides: ln ( 4 ) = ln (( 1.05 ) x )
Using Power Rule of Logarithms Use the power rule of logarithms to get: ln ( 4 ) = x ln ( 1.05 )
Isolating x Solve for x :
x = ln ( 1.05 ) ln ( 4 )
Calculating x Calculate the value of x . The result of the operation is approximately 28.4134. x ≈ 28.4134
Finding the Year Add the value of x to 2000 to find the year when the population reaches 200 million: Y e a r = 2000 + x ≈ 2000 + 28.4134 = 2028.4134
Final Answer The population will reach 200 million in approximately the year 2028.4. Since the question asks for the year, we can say that the population will reach 200 million during the year 2028.
Examples
Exponential growth models, like the one used for the island's population, are commonly used in finance to calculate compound interest. For example, if you invest a certain amount of money at a fixed interest rate, the value of your investment grows exponentially over time. Understanding exponential growth helps you predict future investment values and make informed financial decisions. Similarly, this type of calculation can be used to model the spread of diseases or the growth of social media users.
The population will reach 200 million in the year 2028, calculated using the equation 200 = 50 ( 1.05 ) x . Solving for x gives approximately 28.41, which corresponds to the year 2028 when added to 2000. Therefore, the answer is D. 2028.
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