A population numbers 19,800 organisms initially and grows by [tex]$16.1 \%$[/tex] each year. Suppose [tex]$P$[/tex] represents population, and [tex]$t$[/tex] the number of years of growth. An exponential function to model the population can be written in the form [tex]$P(t)=P_0 \cdot b^t$[/tex]. Give the function below.
Answer (2)
Identify the initial population: P 0 = 19800 .
Calculate the growth factor: b = 1 + 0.161 = 1.161 .
Substitute P 0 and b into the exponential function: P ( t ) = 19800 × ( 1.161 ) t .
The exponential function that models the population growth is P ( t ) = 19800 × ( 1.161 ) t .
Explanation
Understanding the Problem We are given an initial population of 19,800 organisms that grows by 16.1% each year. We need to find an exponential function of the form P ( t ) = P 0 × b t that models this population growth.
Identifying the Initial Population First, we identify the initial population, P 0 , which is given as 19,800.
Calculating the Growth Factor Next, we need to find the base of the exponential function, b . Since the population grows by 16.1% each year, we can calculate b as 1 + growth rate (as a decimal). The growth rate as a decimal is 100 16.1 = 0.161 . Therefore, b = 1 + 0.161 = 1.161 .
Forming the Exponential Function Now, we substitute P 0 = 19800 and b = 1.161 into the exponential function P ( t ) = P 0 × b t to get the function that models the population growth: P ( t ) = 19800 × ( 1.161 ) t .
Examples
Exponential functions are incredibly useful in modeling population growth, but they also have applications in finance. For example, if you invest money in an account that earns compound interest, the growth of your investment can be modeled using an exponential function. Understanding exponential growth can help you make informed decisions about your savings and investments.
The model for the population growth of 19,800 organisms increasing by 16.1% per year is given by the function P ( t ) = 19800 × ( 1.161 ) t . This function represents how the population changes over time. The growth factor is calculated as 1 plus the growth rate expressed as a decimal, resulting in a value of 1.161.
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