To determine the trend of the rat population growth and predict the population on a future day, we analyze the data provided:
Day | Rat Population
:-: | :-:
1 | 4
12 | 12
21 | 35
26 | 62
28 | 86
29 | 107
30 | 115
31 | 132
### Identifying the Trend:
By fitting a second-degree polynomial to the given data points, the equation obtained was:
[tex]\[ P(x) = 0.23737637x^2 - 3.73471918x + 11.33962537 \][/tex]
Here, the leading coefficient of the polynomial (0.23737637) is positive. A positive leading coefficient in a second-degree polynomial indicates that the parabola opens upwards, suggesting an increasing trend in the rat population over the given days.
Fill in the first blank:
The data shows [tex]\( \text{an increasing} \)[/tex] trend.
### Predicting the Population on the 34th Day:
Using the polynomial equation, we substitute [tex]\( x = 34 \)[/tex] to find the rat population on the 34th day:
[tex]\[ P(34) = 0.23737637(34)^2 - 3.73471918(34) + 11.33962537 \approx 158.77 \][/tex]
Therefore, the predicted rat population on the 34th day is approximately 158.77.
Fill in the second blank:
Based on the table, we can assume that the rat population on the [tex]\( 34^{\text{th }} \)[/tex] day will [tex]\( \text{be 158.77} \)[/tex].
So, the completely filled statement would be:
The data shows [tex]\(\text{an increasing}\)[/tex] trend.
Based on the table, we can assume that the rat population on the [tex]\(34^{\text{th}}\)[/tex] day will [tex]\(\text{be 158.77}\)[/tex].
