Question

An insect population is growing such that each new generation is 2.5 times as large as the previous generation. Suppose there were 54 insects in the first generation. The function P(n)=54(2.5)n−1𝑃(𝑛)=54(2.5)𝑛−1 represents the insect population for generation n. Interpret the meaning of P(4)=843.75𝑃(4)=843.75 .(1 point)ResponsesThere were 843 insects in the 4th generation.There were 843 insects in the 4th generation.There were 844 insects in the 4th generation.There were 844 insects in the 4th generation.There were 4 insects in the 844th generation.There were 4 insects in the 844th generation.There were 4 insects in the 843rd generation.

Answer 1

To interpret the meaning of \( P(4) = 843.75 \) in the context of the insect population growth, we first look at what the function \( P(n) = 54(2.5)^{n-1} \) represents.

The function describes the population of insects in generation \( n \), where:

- \( 54 \) is the initial population (when \( n = 1 \)).
- \( 2.5 \) is the growth factor for each generation.

When we evaluate \( P(4) \):

\[
P(4) = 54(2.5)^{4-1} = 54(2.5)^3
\]
Calculating \( (2.5)^3 \):

\[
(2.5)^3 = 15.625
\]

Now, substituting this back into the equation:

\[
P(4) = 54 \times 15.625 = 843.75
\]

The interpretation of \( P(4) = 843.75 \) is that the calculated insect population for the 4th generation is 843.75. Since population counts must be whole numbers, this implies that the population is approximately 844 insects when rounded.

Therefore, the correct interpretation is:

**There were approximately 844 insects in the 4th generation.**