These exercises use the population growth model. Fox...

These exercises use the population growth model. Fox Population The fox population in a certain region has a relative growth rate of $8 \%$ per year. It is estimated that the population in 2013 was $18,000$. (a) Find a function $n(t)=n_{0} e^{n t}$ that models the population t years after 2013 . (b) Use the function from part (a) to estimate the fox population in the year 2021 . (c) After how many years will the fox population reach $25,000 ?$ (d) Sketch a graph of the fox population function for the years $2013-2021 .$

These exercises use the population growth model. Fox Population The fox population in a certain region has a relative growth rate of $8 \%$ per year. It is estimated that the population in 2013 was $18,000$.
(a) Find a function $n(t)=n_{0} e^{n t}$ that models the population t years after 2013 .
(b) Use the function from part (a) to estimate the fox population in the year 2021 .
(c) After how many years will the fox population reach $25,000 ?$
(d) Sketch a graph of the fox population function for the years $2013-2021 .$

Key Concepts

Exponential Growth Model

This model describes how a quantity increases over time at a rate proportional to its current value. It uses the form n(t) = n?e^(rt), where n? is the initial quantity, r is the constant growth rate, and t is time. This framework is commonly applied in population dynamics, finance, and various natural processes.

Relative Growth Rate

The relative growth rate is expressed as a percentage or decimal that indicates the proportional change per unit of time. It directly determines how quickly the quantity grows in the exponential model. An 8% annual growth rate, for instance, corresponds to r = 0.08 in the exponential function.

Solving Exponential Equations

To find the time needed for a quantity to reach a certain level in an exponential model, logarithms are used. By setting the exponential equation equal to the desired value and applying logarithmic properties, one can solve for the time variable efficiently.

Graphing Exponential Functions

Graphing involves plotting the exponential function over a specified time interval. Key features include the initial value, the rate of increase, and the smooth, continuously increasing curve typical of exponential growth. Understanding these characteristics helps in visualizing and interpreting the behavior of the model over time.

Use of the Natural Base 'e'

The constant 'e' (approximately 2.71828) is fundamental to continuous exponential growth models. It serves as the base of the natural logarithm and ensures that the growth process is modeled continuously. Its properties simplify many calculations in growth and decay processes and are integral to the formulation of the exponential growth equation.

Transcript

00:01 Hi.

00:03 So this problem is related to the fox population, the population of foxes in a region.

00:13 And it is given that the population grows by a relative rate of 8 % per year, r equals equal to 8%.

00:29 That is every year it grows 8 % of what it was in that year.

00:36 After that year, it is 8 % more of what it was in that year.

00:44 Okay, it is given that in 2013 the population of foxes was 18 ,000 so we have to we have based on this information we have to find answers to some questions so in part a in part a need to find a function of the number of population with respect to t and so this is the our e to the power r t this is the this is the formula that is if that is if the relative growth rate is given then this is the formula that relates n to number of population with respect to time to time and this n not is the initial population so it is it is said that this model should model should take 2013 as the initial year and we know that in turn 13 the population is 18 ,000 so in our case this would be n -not so put the values of n -not and r in this equation and we'll get the you will get the function in t it is 18 ,000 times e to the power so r is 8 percent 8 percent means 0 .08 0 .08 t okay so this is the solution of the first part part a now what is being asked in part b in part b we have to use this function that that we have got and we have to estimate the population in 2021 so t is equal to not t is not equal to 2021 t is equal to t is equal to the difference between 2021 and 2013 that is the number of years after 2021 the number of years after which the population is asked so this is 8 so at t is equal to 8 here because 2021 is 8 years after 2013 so we will n8 we have to find and this n8 is equal to n0 which is 18 000 times e to the power 0 .08 times 8 which is equal to 8 to the power 0 .64 times 18 ,000.

03:57 Let me calculate this to the power 0 .64 times 18 ,000.

04:12 So the population of foxes in the year is around 3 ,4, 1, 3, 3, 6 foxes...

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