Questions: A population numbers 20,000 organisms initially and decreases by 4.1% each year. Suppose P represents population, and t the number of years of growth. An exponential model for the population can be written in the form P=ab^t, where P=

A population numbers 20,000 organisms initially and decreases by 4.1% each year. Suppose P represents population, and t the number of years of growth. An exponential model for the population can be written in the form P=ab^t, where P=

Transcript text: A population numbers 20,000 organisms initially and decreases by $4.1 \%$ each year. Suppose $P$ represents population, and $t$ the number of years of growth. An exponential model for the population can be written in the form $P=a b^{t}$, where \[ P= \] $\square$

Solution

Solution Steps

Step 1: Identify the initial population (\(P_0\))

The initial population size is given as \(P_0 = 20000\).

Step 2: Determine the rate of decrease (\(r\))

The rate of decrease per time period is given as \(r = 0.041\), expressed as a decimal.

Step 3: Calculate the base of the exponential function (\(b\))

Using the formula \(b = 1 - r\), we find the base of the exponential function to be \(b = 0.959\).

Step 4: Model the population change

Using the exponential decay formula \(P = P_0 \cdot b^t\), we model the population over time \(t\).

Step 5: Predict future population sizes

For a given time \(t = 5\), the predicted population size is \(P = 16222.7\).

Final Answer:

The predicted population size after \(t = 5\) time periods is \(P = 16222.7\), rounded to 2 decimal places.

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