A population numbers 20,000 organisms initially and decreases by 11.7% each year. Suppose \(P\) represents the population, and \(t\) the number of years of decline.

An exponential model for the population can be represented by which of the following equations?

1) \[P = 20000 - 11.7t\]

2) \[P = 20000 + 11.7t\]

3) \[P = 20000 \times (0.883)^t\]

4) \[P = 20000 / 11.7t\]

None of the provided options accurately represent an exponential model for the decline in population. The correct model for exponential decay of a population is P(t) = P(0) * e^{-rt}, which is not listed among the choices given.Hence,option 1 is correct,P = 20000 - 11.7t.

The student is asking for an exponential model of a declining population. An exponential decrease in population would be represented by the equation P(t) = P(0) imes e^{-rt}, where P(t) is the population at time t, P(0) is the initial population, e is the base of the natural logarithm, and r is the rate of decline. Given that the initial population is 20,000 organisms and it decreases by 11.7% per year, the correct model would be P = 20000 imes e^{-0.117t}.

The options presented do not include an exponential decay function, so it can be assumed that there is a typo or an error in the provided choices. The actual equation should include the exponential function e to properly model the exponential decay.

Hence,option 1 is correct,P = 20000 - 11.7t.