A Population Of 2500 Frogs Increases Exponentially At An Annual Rate Of $25\%$. Which Equation Below Best Represents The Population ($P$) In $t$ Years?A. $P = 2500 + 0.25t$ B. $P = 2500(0.25)^t$ C. $P

Feb 27, 2025 by ADMIN 203 views

A Population of 2500 Frogs: Exponential Growth and Modeling

In the natural world, populations of living organisms can grow or decline at varying rates. Understanding these growth patterns is crucial for predicting population sizes, managing resources, and making informed decisions about conservation and management. In this article, we will explore the concept of exponential growth and model a population of 2500 frogs that increases at an annual rate of 25%.

Exponential growth occurs when a quantity increases by a fixed percentage over a fixed period. This type of growth is characterized by a rapid increase in the population size, with the rate of growth accelerating over time. The formula for exponential growth is given by:

P(t) = P0 * (1 + r)^t

where:

P(t) is the population size at time t
P0 is the initial population size
r is the annual growth rate (expressed as a decimal)
t is the time in years

Given that the population of 2500 frogs increases exponentially at an annual rate of 25%, we can use the formula for exponential growth to model the population size over time. The initial population size is 2500, and the annual growth rate is 25% or 0.25 as a decimal.

Option A: P = 2500 + 0.25t

This option represents a linear growth model, where the population size increases by a fixed amount each year. However, this is not an exponential growth model, as the population size does not increase by a fixed percentage over time.

Option B: P = 2500(0.25)^t

This option represents an exponential growth model, where the population size increases by a fixed percentage (25%) over time. The formula is in the correct form, with the initial population size (2500) multiplied by the growth factor (0.25) raised to the power of time (t).

Option C: P = 2500e^(0.25t)

This option represents an exponential growth model, but with a different growth factor. The formula uses the natural exponential function (e) and the growth rate (0.25) as an exponent. While this is a valid exponential growth model, it is not the best representation of the population growth in this scenario.

Based on the given information, the equation that best represents the population of 2500 frogs in t years is:

P = 2500(0.25)^t

This equation accurately models the exponential growth of the population, with the population size increasing by a fixed percentage (25%) over time. The formula is in the correct form, with the initial population size (2500) multiplied by the growth factor (0.25) raised to the power of time (t).

Exponential growth models have numerous applications in various fields, including:

Biology: Modeling population growth and decline in ecosystems
Economics: Predicting economic growth and inflation
Finance: Calculating compound interest and investment returns
Environmental Science: Modeling the spread of diseases and pollutants

In conclusion, understanding exponential growth and modeling population sizes is crucial for making informed decisions about conservation, management, and resource allocation. By applying the correct formula and growth rate, we can accurately predict population sizes and make data-driven decisions.
A Population of 2500 Frogs: Exponential Growth and Modeling - Q&A

In our previous article, we explored the concept of exponential growth and modeled a population of 2500 frogs that increases at an annual rate of 25%. We also discussed the different options for representing the population growth and concluded that the equation P = 2500(0.25)^t is the best representation of the population growth in this scenario.

Q: What is exponential growth?

A: Exponential growth occurs when a quantity increases by a fixed percentage over a fixed period. This type of growth is characterized by a rapid increase in the population size, with the rate of growth accelerating over time.

Q: What is the formula for exponential growth?

A: The formula for exponential growth is given by:

P(t) = P0 * (1 + r)^t

where:

P(t) is the population size at time t
P0 is the initial population size
r is the annual growth rate (expressed as a decimal)
t is the time in years

Q: What is the difference between exponential growth and linear growth?

A: Exponential growth occurs when a quantity increases by a fixed percentage over a fixed period, whereas linear growth occurs when a quantity increases by a fixed amount each year.

Q: How do I calculate the population size using the exponential growth formula?

A: To calculate the population size using the exponential growth formula, you need to know the initial population size (P0), the annual growth rate (r), and the time (t). You can then plug these values into the formula:

P(t) = P0 * (1 + r)^t

Q: What is the significance of the growth rate (r) in the exponential growth formula?

A: The growth rate (r) determines the rate at which the population size increases. A higher growth rate will result in a faster increase in population size, while a lower growth rate will result in a slower increase.

Q: Can I use the exponential growth formula to model population decline?

A: Yes, you can use the exponential growth formula to model population decline by using a negative growth rate (r). For example, if the population is declining at an annual rate of 10%, you can use the formula:

P(t) = P0 * (1 - 0.10)^t

Q: What are some real-world applications of exponential growth?

A: Exponential growth models have numerous applications in various fields, including:

Biology: Modeling population growth and decline in ecosystems
Economics: Predicting economic growth and inflation
Finance: Calculating compound interest and investment returns
Environmental Science: Modeling the spread of diseases and pollutants

For further reading and exploration, we recommend the following resources:

Mathematics textbooks: Exponential growth and modeling are covered in most mathematics textbooks, including algebra and calculus.
Online resources: Websites such as Khan Academy, Wolfram Alpha, and Mathway offer interactive tutorials and examples of exponential growth and modeling.
Scientific journals: Scientific journals such as the Journal of Mathematical Biology and the Journal of Theoretical Biology publish research on exponential growth and modeling in various fields.