Question

The population of a certain substance begins at 500 and decreases by 2.5% every day. Write a function to represent the population at time t = 0, with t = the number of days after the initial day.​

Answer 1

The population function that models the decrease of a substance by 2.5% daily is P(t) = 500 e-ln(1.025)t, where P(t) is the population at any given day t after the initial measurement.

Step-by-step explanation:

The student is asking for a function that models the population of a substance that decreases by 2.5% every day, starting from an initial population of 500. To write this function, we need to use an exponential decay formula. The general form of an exponential decay function is N(t) = N0 e-kt, where N(t) is the population at time t, N0 is the initial population, and k is the decay constant.

Since the population decreases by 2.5% each day, we can convert the percentage to a decimal, which gives 0.025. The decay constant k is then the natural log of 1 minus the decay percentage, which is ln(1 - 0.025). The resulting function to model the population at time t is:

Population Function: P(t) = 500 e-ln(1.025)t

This function allows us to calculate the population for any given day t after the initial measurement.