Thank you for visiting Which of the following equations would model a population with an initial size of 625 that is growing at an annual rate of tex 8. 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 :
To find the correct equation modeling the population growth, let's break down the problem:
1. Understand the Problem: We need an equation that represents a population with an initial size of 625, growing annually at a rate of 8.5%. This is a typical scenario for exponential growth.
2. Exponential Growth Formula: In exponential growth, the population at any time [tex]\( t \)[/tex] can be modeled by the formula:
[tex]\[
P = P_0 \times (1 + r)^t
\][/tex]
where:
- [tex]\( P \)[/tex] is the population after time [tex]\( t \)[/tex],
- [tex]\( P_0 \)[/tex] is the initial population,
- [tex]\( r \)[/tex] is the growth rate (as a decimal),
- [tex]\( t \)[/tex] is the time in years.
3. Apply the Values:
- The initial population [tex]\( P_0 \)[/tex] is 625.
- The growth rate [tex]\( r \)[/tex] is 8.5%, which is 0.085 in decimal form.
4. Substitute into the Formula:
[tex]\[
P = 625 \times (1 + 0.085)^t
\][/tex]
Simplifying:
[tex]\[
P = 625 \times (1.085)^t
\][/tex]
5. Match with the Options:
- (1) [tex]\( P = 625 \times (8.5)^t \)[/tex] is incorrect because it uses the growth factor 8.5 instead of 1.085.
- (2) [tex]\( P = 625 \times (1.085)^t \)[/tex] is the correct representation as it matches our derived formula.
- (3) [tex]\( P = 1.085^t + 625 \)[/tex] does not follow the exponential growth formula.
- (4) [tex]\( P = 8.5 t^2 + 625 \)[/tex] is not an exponential model and incorrectly applies the growth concept.
Therefore, the equation that correctly models the population growth is option (2): [tex]\( P = 625 \times (1.085)^t \)[/tex].
1. Understand the Problem: We need an equation that represents a population with an initial size of 625, growing annually at a rate of 8.5%. This is a typical scenario for exponential growth.
2. Exponential Growth Formula: In exponential growth, the population at any time [tex]\( t \)[/tex] can be modeled by the formula:
[tex]\[
P = P_0 \times (1 + r)^t
\][/tex]
where:
- [tex]\( P \)[/tex] is the population after time [tex]\( t \)[/tex],
- [tex]\( P_0 \)[/tex] is the initial population,
- [tex]\( r \)[/tex] is the growth rate (as a decimal),
- [tex]\( t \)[/tex] is the time in years.
3. Apply the Values:
- The initial population [tex]\( P_0 \)[/tex] is 625.
- The growth rate [tex]\( r \)[/tex] is 8.5%, which is 0.085 in decimal form.
4. Substitute into the Formula:
[tex]\[
P = 625 \times (1 + 0.085)^t
\][/tex]
Simplifying:
[tex]\[
P = 625 \times (1.085)^t
\][/tex]
5. Match with the Options:
- (1) [tex]\( P = 625 \times (8.5)^t \)[/tex] is incorrect because it uses the growth factor 8.5 instead of 1.085.
- (2) [tex]\( P = 625 \times (1.085)^t \)[/tex] is the correct representation as it matches our derived formula.
- (3) [tex]\( P = 1.085^t + 625 \)[/tex] does not follow the exponential growth formula.
- (4) [tex]\( P = 8.5 t^2 + 625 \)[/tex] is not an exponential model and incorrectly applies the growth concept.
Therefore, the equation that correctly models the population growth is option (2): [tex]\( P = 625 \times (1.085)^t \)[/tex].
Thank you for reading the article Which of the following equations would model a population with an initial size of 625 that is growing at an annual rate of tex 8. 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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