Answer:
Step-by-step explanation:
To find the average rate of change of the population from 2010 to 2015 using the given model \( P(t) = 700(0.992)^t \), we need to calculate the population at the two given times and then find the average rate of change over that interval.
First, let's find the population at \( t = 0 \) (2010) and \( t = 5 \) (2015).
At \( t = 0 \):
\[ P(0) = 700(0.992)^0 = 700 \]
At \( t = 5 \):
\[ P(5) = 700(0.992)^5 \]
Now, let's calculate \( P(5) \):
\[ P(5) = 700(0.992)^5 \]
\[ P(5) = 700(0.992)^2 \]
\[ P(5) = 700(0.984064) \]
\[ P(5) ≈ 688.844 \]
Now, we can find the average rate of change using the formula:
\[ \text{Average rate of change} = \frac{\text{Change in population}}{\text{Change in time}} \]
\[ \text{Average rate of change} = \frac{P(5) - P(0)}{5 - 0} \]
\[ \text{Average rate of change} = \frac{688.844 - 700}{5} \]
\[ \text{Average rate of change} = \frac{-11.156}{5} \]
\[ \text{Average rate of change} ≈ -2.2312 \]
Therefore, the average rate of change from 2010 to 2015 is approximately \(-2.2312\) thousand per year.
