Thank you for visiting An initial population of 625 quail increases at an annual rate of tex 14 tex Write an exponential function to model the quail population A. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
We start with an initial quail population of
[tex]$$
625.
$$[/tex]
The annual rate of increase is given as 14%. This means that each year the population grows by 14% of the current number of quail. To find the factor by which the population increases each year, we add 1 (representing the entire current population) to the growth rate (expressed as a decimal):
[tex]$$
1 + 0.14 = 1.14.
$$[/tex]
This factor, 1.14, tells us that each year the quail population is multiplied by 1.14.
Thus, the exponential function that models the quail population after [tex]$x$[/tex] years is written as:
[tex]$$
f(x) = 625 \cdot (1.14)^x.
$$[/tex]
Here, [tex]$625$[/tex] is the initial population and [tex]$1.14^x$[/tex] represents the multiplicative effect of the 14% annual increase over [tex]$x$[/tex] years.
To confirm the behavior of the model:
- At [tex]$x = 0$[/tex], the population is
[tex]$$
f(0) = 625 \cdot (1.14)^0 = 625 \cdot 1 = 625,
$$[/tex]
as expected.
- At [tex]$x = 1$[/tex], the population is
[tex]$$
f(1) = 625 \cdot (1.14) \approx 712.5,
$$[/tex]
which matches the computed value.
Therefore, the correct exponential function to model the quail population is
[tex]$$
\boxed{f(x) = 625 \cdot (1.14)^x.}
$$[/tex]
[tex]$$
625.
$$[/tex]
The annual rate of increase is given as 14%. This means that each year the population grows by 14% of the current number of quail. To find the factor by which the population increases each year, we add 1 (representing the entire current population) to the growth rate (expressed as a decimal):
[tex]$$
1 + 0.14 = 1.14.
$$[/tex]
This factor, 1.14, tells us that each year the quail population is multiplied by 1.14.
Thus, the exponential function that models the quail population after [tex]$x$[/tex] years is written as:
[tex]$$
f(x) = 625 \cdot (1.14)^x.
$$[/tex]
Here, [tex]$625$[/tex] is the initial population and [tex]$1.14^x$[/tex] represents the multiplicative effect of the 14% annual increase over [tex]$x$[/tex] years.
To confirm the behavior of the model:
- At [tex]$x = 0$[/tex], the population is
[tex]$$
f(0) = 625 \cdot (1.14)^0 = 625 \cdot 1 = 625,
$$[/tex]
as expected.
- At [tex]$x = 1$[/tex], the population is
[tex]$$
f(1) = 625 \cdot (1.14) \approx 712.5,
$$[/tex]
which matches the computed value.
Therefore, the correct exponential function to model the quail population is
[tex]$$
\boxed{f(x) = 625 \cdot (1.14)^x.}
$$[/tex]
Thank you for reading the article An initial population of 625 quail increases at an annual rate of tex 14 tex Write an exponential function to model the quail population A. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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Rewritten by : Jeany
