At time 𝑡=0 years, the population of a certain city was 23,144. During each of the next 10 years, the population decreased by 4% per year. Based on this information, which of the following models the population as a function of time 𝑡, in years, for 0≤𝑡≤10 ?Responses23,144-0.04𝑡23,144 minus 0.04 t23,144-0.96𝑡23,144 minus 0.96 t23,1440.96𝑡23,144 open parentheses 0.96 close parentheses to the power of t
Question
At time 𝑡=0 years, the population of a certain city was 23,144. During each of the next 10 years, the population decreased by 4% per year. Based on this information, which of the following models the population as a function of time 𝑡, in years, for 0≤𝑡≤10 ?Responses23,144-0.04𝑡23,144 minus 0.04 t23,144-0.96𝑡23,144 minus 0.96 t23,1440.96𝑡23,144 open parentheses 0.96 close parentheses to the power of t
Solution
The correct model for the population as a function of time 𝑡, in years, for 0≤𝑡≤10 is 23,144(0.96)^𝑡.
Here's why:
The population is decreasing by 4% each year. This means that each year, the population is 96% of what it was the previous year (since 100% - 4% = 96%).
So, if we start with a population of 23,144, after one year the population would be 23,144 * 0.96. After two years, the population would be 23,144 * 0.96 * 0.96, and so on.
This can be written more succinctly as 23,144 * (0.96)^t, where t is the number of years that have passed. This is an example of an exponential decay model.
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