If a population grows by 3 percent each year, the growth of the population is
A. exponential
B. cubic
C. logarithmic
D. linear
Answer (2)
A population that grows by 3 percent each year exhibits exponential growth, as it can be mathematically described by the equation P t = P 0 ( 1.03 ) t . This indicates that the growth rate is proportional to the current population size. Hence, the correct answer is \textbf{exponential}.
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The population increases by 3% each year.
After t years, the population is P t = P 0 ( 1.03 ) t .
The population P t is an exponential function of time t .
Therefore, the growth is exponential .
Explanation
Problem Analysis Let's analyze the problem. We are given that a population grows by 3 percent each year. We need to determine the type of growth this represents.
Calculating Population Growth Let P 0 be the initial population. After one year, the population is P 1 = P 0 + 0.03 P 0 = P 0 ( 1 + 0.03 ) = 1.03 P 0 . After two years, the population is P 2 = P 1 ( 1.03 ) = P 0 ( 1.03 ) 2 . After t years, the population is P t = P 0 ( 1.03 ) t .
Determining the Type of Growth The population P t is an exponential function of time t , since it is of the form P t = P 0 × ( 1.03 ) t , where the variable t is in the exponent. Therefore, the growth is exponential.
Final Answer The growth of the population is exponential.
Examples
Exponential growth is a pattern of data that shows greater increases with passing time, creating the curve of an exponential function. A real-world example is the spread of a virus. If one person has a virus and infects two people, and each of those two people infects two more, the number of newly infected people grows exponentially. This is because the number of people who can spread the virus increases with each generation, leading to rapid growth. Another example is compound interest, where the amount of interest earned increases over time as the principal grows.
