To solve the problem of modeling a population that starts at 10,000 organisms and decreases by 7.4% each year using an exponential function, we can proceed step-by-step as follows:
1. Understand the Initial Population and the Decay Rate:
- Initial population (\(P_0\)) = 10,000 organisms
- Decay rate = 7.4%
2. Convert the Decay Rate to Decimal Form:
- Decay rate in decimal form (\(r\)) = \(-7.4\%\) = \(-\frac{7.4}{100}\) = \(-0.074\)
3. Formulate the General Exponential Model:
- The general exponential form is \(P = a b^t\), where
- \(a\) is the initial amount or population.
- \(b\) is the base of the exponential function.
- \(t\) is the time in years.
4. Define the Constants in the Model:
- \(a\) = Initial population = 10,000
- To find \(b\):
- The population decreases by 7.4% each year, meaning it retains \(100\% - 7.4\%\) of the population each year.
- Hence, \(b = 1 - 0.074 = 0.926\)
5. Consolidate the Model:
- By substituting these values into the general form, we get:
- \(P = 10000 \times 0.926^t\)
So, the exponential model for the population after \( t \) years is:
[tex]\[
P = 10000 \times 0.926^t
\][/tex]
This completes the formulation of the problem.
