Questions: Suppose some flour beetles are left undisturbed in a warehouse bin. The function P(x) below can be used to determine the number of beetles, P, after X weeks. Based on this function, what is true about the average rate of change between 3 and 5 weeks? P(x)=25(1.2)^x The population is decreasing by approximately 10 beetles per week The population is increasing by approximately 10 weeks per beetle The population is increasing by approximately 10 beetles per week The population is decreasing by approximately 10 weeks per beetle.

Solution Steps

The function given is \( P(x) = 25(1.2)^x \), where \( P(x) \) represents the population of beetles after \( x \) weeks. We are asked to determine the average rate of change of the population between 3 and 5 weeks.

First, calculate the population at \( x = 3 \) weeks: \[ P(3) = 25(1.2)^3 = 25(1.728) = 43.2 \]

Next, calculate the population at \( x = 5 \) weeks: \[ P(5) = 25(1.2)^5 = 25(2.48832) = 62.208 \]

The average rate of change between \( x = 3 \) and \( x = 5 \) is given by: \[ \text{Average rate of change} = \frac{P(5) - P(3)}{5 - 3} = \frac{62.208 - 43.2}{2} = \frac{19.008}{2} = 9.504 \]

The average rate of change is approximately 9.504 beetles per week. This means the population is increasing by approximately 9.504 beetles per week between 3 and 5 weeks.

Final Answer