For the following exercises, consider this scenario: For...

For the following exercises, consider this scenario: For each year $t,$ the population of a forest of trees is represented by the function $A(t)=115(1.025)^{t} .$ In a neighboring forest, the population of the same type of tree is represented by the function $B(t)=82(1.029)^{t} .$ (Round answers to the nearest whole number.) Discuss the above results from the previous four exercises. Assuming the population growth models continue to represent the growth of the forests, which forest will have the greater number of trees in the long run? Why? What are some factors that might influence the long-term validity of the exponential growth model?

For the following exercises, consider this scenario: For each year $t,$ the population of a forest of trees is represented by the function $A(t)=115(1.025)^{t} .$ In a neighboring forest, the population of the same type of tree is represented by the function $B(t)=82(1.029)^{t} .$ (Round answers to the nearest whole number.)
Discuss the above results from the previous four exercises. Assuming the population growth models continue to represent the growth of the forests, which forest will have the greater number of trees in the long run? Why? What are some factors that might influence the long-term validity of the exponential growth model?

Key Concepts

Long-run Behavior of Exponential Functions

Exponential functions with constant growth rates eventually dominate because their outputs increase multiplicatively over time. This means that small differences in growth rates can result in significant differences in outcomes over the long run, often enabling a system with a higher growth factor to surpass another with a lower factor despite initial disparities. This principle is directly applicable in determining which forest will eventually have a greater population.

Model Assumptions and Limitations

Exponential growth models assume unlimited resources and constant growth rates, conditions that rarely hold indefinitely in real-world scenarios. Environmental factors, competition for resources, disease, and carrying capacities can alter or limit growth, potentially making the model less accurate over time. Understanding these assumptions and limitations is crucial when applying such models to long-term predictions as real-world systems may experience slowdown or stabilization.

Exponential Growth Model

This model describes processes where the rate of change of a quantity is proportional to its current value. In the context of population growth, it means that the forest’s number of trees increases by a fixed percentage over equal time intervals, resulting in a multiplicative growth pattern. The formula A(t)=initial value*(growth factor)^t captures this behavior with t representing time, initial value as the starting population, and the growth factor indicating the percentage increase per time unit.

Relative Growth Rate

The relative growth rate in an exponential model is determined by the growth factor, where a higher factor indicates a faster increase per period. Even if one forest starts with a smaller initial population, a slightly larger growth rate (as reflected by a larger multiplier in its exponential function) means the population will eventually exceed the one with a lower rate. This concept is key in comparing long-term behavior of different systems modeled by exponential functions.

Transcript

00:02 Right.

00:03 So for each year, the population of forest trees is represented by these two functions, a and b.

00:12 And discuss the above results from the pre -use four exercises and assuming the population growth models continue to represent the growth of the forests.

00:20 Which forests will have the greater number of trees in the long run and why? and what are some factors that might influence the validity of this growth model? so pretty similar to what we've already done in the past.

00:33 With the same type of question.

00:35 Basically, we have these two equations, right? we have 115 times 1 .025 to the t power.

00:41 We have 82 times 1 .029 to the t power.

00:44 Now, one thing that's interesting is a lot of people usually get mistaken on which will grow faster and increase more in the long run.

00:55 Because by the looks of it, right, you would think it's a.

00:58 Because a has not only a really big number right over here and be multiplied by, you know, one 0 .025 to the t versus b which is much smaller it's only 82 times you know basically the same number right it's 1 .029 it's like a 0 .0 4 oh 4 difference as compared to 1 .025 so you think that hey you know they're basically the same number so mainly just matters on this number right over here and while this is partially true in the beginning in the long run not so much and i'll tell you why in just a second but first we have to know what this means exactly what what are the components of those equation and what do they represent exactly? so the number right over here is typically the starting population.

01:42 So in this case for a, it started off with 115 trees, and then every year it increases by a rate of 1 .025.

01:51 So every single year, you have 115 times for the first year, it would be 115 times 1 .025, which would give you a slightly bigger number, right? 1 .15 times 0 .025 is going to be about equal to 118.

02:17 And then you can see that the next year it would be 1 .025 squared.

02:22 And then that would be a slightly bigger number.

02:25 And it would go on and on and on.

02:27 While for b, it starts off at 100, not 100, at 82, and it's going to increase by a rate of 1 .029.

02:35 Every single year.

02:38 So it's actually a slightly higher rate for b...

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