To interpret the function \( p(n) = 54 \times 2.5^{n-1} \) for the insect population, let's break it down:
- \( p(n) \) represents the population of insects in the \( n \)-th generation.
- The initial population (first generation, or \( n = 1 \)) is 54 insects.
- Each subsequent generation has a population that is 2.5 times larger than the previous generation. This is reflected in the term \( 2.5^{n-1} \).
Now, let's calculate \( p(4) \):
\[ p(4) = 54 \times 2.5^{4-1} = 54 \times 2.5^3 \]
First, calculate \( 2.5^3 \):
\[ 2.5^3 = 2.5 \times 2.5 \times 2.5 = 6.25 \times 2.5 = 15.625 \]
Now substitute this back into the equation for \( p(4) \):
\[ p(4) = 54 \times 15.625 = 843.75 \]
So, \( p(4) = 843.75 \).
Interpretation:
The value \( p(4) = 843.75 \) indicates that in the fourth generation of this insect population, there will be approximately 843.75 insects. Since insect populations are typically whole numbers, this value can be understood to mean that the population is growing rapidly and that by the fourth generation, the population would have substantially increased from the initial 54 insects. The fractional part suggests that if the growth continues, we can anticipate reaching 844 insects soon in terms of rounding, or it indicates the growth model's theoretical prediction rather than an exact count.