To determine the average rate of change of the fruit fly population over the interval [tex]\([0, 10.5]\)[/tex], we'll use the given population model [tex]\(P=10(1.27)^t\)[/tex].
Here's the step-by-step solution:
1. Identify the initial and final times in the given interval:
[tex]\[
t_{\text{initial}} = 0, \quad t_{\text{final}} = 10.5
\][/tex]
2. Calculate the population at the initial time ([tex]\(t = 0\)[/tex]):
[tex]\[
P_{\text{initial}} = 10 \times (1.27)^0 = 10 \times 1 = 10
\][/tex]
3. Calculate the population at the final time ([tex]\(t = 10.5\)[/tex]):
[tex]\[
P_{\text{final}} = 10 \times (1.27)^{10.5} \approx 123.01
\][/tex]
4. Determine the average rate of change:
[tex]\[
\text{Average rate of change} = \frac{P_{\text{final}} - P_{\text{initial}}}{t_{\text{final}} - t_{\text{initial}}}
\][/tex]
Plugging in the values:
[tex]\[
\text{Average rate of change} = \frac{123.01 - 10}{10.5 - 0} = \frac{113.01}{10.5} \approx 10.76
\][/tex]
5. Include appropriate units: Since [tex]\(P\)[/tex] represents the number of fruit flies and [tex]\(t\)[/tex] is in days, the average rate of change is in fruit flies per day.
Thus, the average rate of change of the population over the interval [tex]\([0, 10.5]\)[/tex] is approximately [tex]\(10.76\)[/tex] fruit flies per day.
