Question

Nstructions:

View the video found on page 1 of this journal activity.
Using the information provided in the video, answer the questions below.
Show your work for all calculations.
Here are the data for the hypothetical (or imagined) rabbit population.

Year (x) 0 1 2 3 4 5 6 7 8 9 10
Number of rabbits (y) 6 8 10 14 19 25 33 43 57 77 104

1. Looking at the data, does this rabbit growth look linear, quadratic, or exponential? Explain your answer.
(1 point)






2. Make a scatterplot for the data. (2 points)

3. Looking at the scatterplot, which regression model do you think fits the data? Explain your answer.
(1 point)






4. Using a graphing calculator, find the quadratic regression equation for the rabbit data.
(2 points: 1 point for identifying constants, 1 for the equation)






5. Using the quadratic regression equation from question 4, predict the number of rabbits in year 70.
(2 points: 1 point for setting up the correct equation, 1 for simplifying)






6. Using a graphing calculator, find the exponential regression equation for the rabbit data.
(2 points: 1 point for identifying the constants, 1 for the equation)






7. Using the exponential regression equation from question 6, predict the number of rabbits in year 70.
(2 points: 1 point for setting up the correct equation, 1 for simplifying)






8. Compare the predicted number of rabbits for year 70 using the quadratic model (question 5) with the predicted number using the exponential model (question 7). Which prediction is larger? (1 point)






9. Is it reasonable to use this data set to predict the hypothetical number of rabbits for year 70? Explain.
(1 point)








Making a Decision

10. These graphs show the scatterplot with the quadratic regression equation and the exponential regression equation.
a) Which do you think is a better fit? Why? (2 points: 1 point for choice, 1 point for explanation)








b) How does your answer to part (a) compare with your guess about the best model in question 1? Was your initial choice correct? (1 point)








Analyze Your Results

11. The video says there were an estimated 10 billion rabbits in Australia after 70 years.
a) How does your predicted number of rabbits for the hypothetical population compare with the actual rabbit population in Australia in year 70? (1 point)








b) Why do you think the rabbit population in Australia can grow so large? (1 point)






c) What are some factors that might prevent a rabbit population from growing infinitely large? (1 point)




PROMPT
Rabbits are cute, aren't they? [An image of a rabbit appears.] But they can have lots of cute babies, and it's not long before the babies start having babies . . .

[A green image of Australia appears.]

In 1859, an Australian had 24 wild rabbits shipped to the land down under. [Small rabbits begin to pop up on the image of Australia and cover it entirely.] Within 70 years, there were an estimated 10 billion rabbits in Australia. That's billion . . . with a B. It's an ecological disaster they're still dealing with.

So, how do we model growth like this mathematically? Here's some hypothetical data that grow a lot like the Australian rabbit population did. [A table showing the population of rabbits per year is shown]. The rabbit population is getting larger and larger, but is this growth linear? Or would a different model work better?

Answer 1

14.3.4 Journal: Nonlinear Models

1.We can rule out a linear trend based on the fact that the population increased from 77 to 104 in just one year. This quick increase alone would indicate that the rabbit population is growing exponentially.

2.

3.The data had previously been recognized as exponential, and a check at the scatterplot would only confirm this. As a result, we will need to utilise the exponential regression equation to appropriately describe the data.

4. The constants: y=Ax^2+Bx+c

The Equation: y=1.2040x^2-3.1396x+9.5594

5.y=1.2040(70)^2-3.1396(70)+9.5594=5689

6.y=5.9385*1.3296^x

7.y=5.9385*1.3296^70=2,717,360,209

8. The prediction in question 7

9.Yes. As long as the data was successfully collected, we have enough points to determine the likely trend of the variable as well as its possible values in the next years. While more data points would increase the prediction, our existing dataset is sufficient.

10a. The exponential model.

b.My answer in question 1 is the same as the previous answer. So I can say that i was correct.

11 a. After 70 years, our response suggests that there will be over 3 billion rabbits. While this is relatively low in comparison to the video's predicted 10 billion rabbits, it is closer than the quadratic model's prediction of 5,689 rabbits.

b. Because each offspring is bound to bring one or more rabbits into the population, the rabbit population in Australia increased enormously.

c.The environment and everyone surrounding the rabbits can prevent their number from expanding indefinitely.

Explanation:

Good luck : )

Answer 2

1. For us to know the answer to this, we just need to examine the table of data. Looking at it, we can rule out the linear trend since the growth of the population rose from 77 to 104 in just one year. This sudden spike alone would tell us that the trend of the growth of the rabbit's population is exponential.

2. To make the scatterplot of the data, we just need to plot the given points in a coordinate plane where the number of rabbits is in the y-axis and the year is in the x-axis. For this problem though, I do not need to attach a plot anymore since you have already given it in your attachments. The scatterplot would just be what you have attached minus the red lines that were used to model the equation (since that is for a later item).

3. We have previously identified the data to be exponential, and looking at the scatterplot would just confirm this. Therefore, for us to model the data accurately, we would need to make use of the exponential regression equation.

4. For this item I just used a calculator online. For the quadratic equation, we just need to find the constants in the equation
`y=A x^(2) +Bx+C`. Substituting the data into the calculator we get the equation:
`y=1.2040x^(2) -3.1396x+9.5594`.

5. In here, the problem is just telling us to use the equation in item #4 to calculate or estimate the number of rabbits in 70 years. Therefore, we just substitute the value (70) to the variable x (in the previous equation, x represents the year while y represents the number of rabbits).


`y=1.2040(70)^(2) -3.1396(70)+9.5594=5689`

6. I used the same calculator I did in item #4 for this item. In an exponential regression equation we just need to find the constants in the equation
`y=A* B^(x)`. According to the calculator, the exponential regression equation is
`y=5.9385* 1.3296^(x)`

7. For this item, we do the same thing that we did in item #5 but this time we substitute the value to the equation in item #6. Since the problem is still asking us to predict the number of rabbits in 70 years, we substitute this value again to x.


`y=5.9385* 1.3296^(70)=2,717,360,209`

8. This question is self-explanatory and we are only being tasked to look at our answers in items #5 and #7 to compare which one of them is larger. Clearly, our prediction in item #7 significantly exceeds our prediction in item #5, therefore this is our larger prediction.

9. Yes, of course. As long as the data was gathered correctly, we have enough points that will allow us to determine the possible trend of the variable as well as its possible values in the future years. While adding more points would improve the prediction, our current dataset would suffice.

10a. By looking at the graphs that you attached, we can clearly see that the exponential model follows all data points. While the quadratic model still hits most points and could even be regarded accurate, its different trend would lead us to infer that the model will go off course in the future values.

10b. Our answer in item #10a is actually the same as our answer in item #1. So in a way, we can confidently say that our initial choice was correct. The similarity in the answers proves that our reasoning in the first item and the previous item support each other.

11a. Our answer predicts that there will almost be 3 billion rabbits after 70 years. While this is relatively low compared to the estimated 10 billion rabbits in the video, this is closer than the 5,689 rabbits estimated by the quadratic model.

11b. The rabbit population in Australia grew exponentially because each offspring is bound to introduce one or more rabbits into the population. This creates endless and fast-growing branches in a family tree that will only go faster as the rabbits grow in number. Without anything to stop them (since they were not made for that environment in the first place. i.e. no competitors), they can grow uninterrupted.

11c. The environment and everyone around the rabbits can hinder their population from growing infinitely large. A most effective hindrance would be the introduction of a competitor or a predator. With someone competing for resources or actively hunting them down, their population would surely see a stop/ a decline. Other factors include diseases, food, natural disasters, etc.