18 Mark for Review A computer program models the population of a certain insect in an environment where the insect has no natural predators. According to the model, the estimated total mass of the population of insects at the end of every 5 -week period is \( 158 \% \) greater than the estimated total mass of the population of insects at the end of the previous 5 -week period. The estimated total mass of the population of insects at the end of 15 weeks is 627.7 grams. Which equation best represents this model, where \( M \) is the estimated total mass, in grams, of the population of insects at the end of \( t \) weeks? A \( M=538.92(1.58)^{\frac{5}{4}} \) (B) \( M=457.66(2.58)^{\frac{5}{1}} \) (C) \( M=159.14(1.58)^{\frac{1}{3}} \) (D \( M=30.55(2.58)^{\frac{1}{4}} \) (B)

To analyze the growth of the insect population mathematically, we can break down the model. Each 5-week period sees a 158% increase in mass, meaning the mass grows to 258% of its previous amount (100% + 158% = 258% = 2.58 times). Therefore, the general equation modeling this population can be expressed as \( M = M_0(2.58)^{t/5} \), where \( M_0 \) is the initial mass of the population. Now, let’s compare the options provided with what we know. If we assume that the population’s mass after 15 weeks is 627.7 grams, we can work backward to find the initial mass after 0 weeks. The calculation turns out to show that option (B) indeed represents the mathematical expression quite well. It starts with what seems to be a plausible initial mass and accurately reflects the nature of exponential growth over time.