To model the decline of a population with a consistent percentage reduction, we can use the exponential decay formula:
\[ f(t) = P_0 \cdot (1 - r)^t \]
Where:
- \( P_0 \) is the initial population,
- \( r \) is the rate of decline (as a decimal),
- \( t \) is the time in years.
In this case:
- The initial population \( P_0 \) is 82.
- The rate of decline \( r \) is 6%, which in decimal form is 0.06.
Since the population is declining, we can express the model as:
\[ f(t) = 82 \cdot (1 - 0.06)^t \] \[ f(t) = 82 \cdot (0.94)^t \]
Now, comparing this with the options you provided:
- Option #1: \( f(x) = 82(1.06)^x \) (This represents growth, not decline)
- Option #2: \( f(x) = 82(1.6)^x \) (This also represents growth)
- Option #3: \( f(x) = 82(0.6)^x \) (This represents a steeper decline, not the correct decay rate)
- Option #4: \( f(x) = 82(0.94)^x \) (This correctly represents the decline at 6% per year)
Thus, the correct option that models the decline is Option #4: \( f(x) = 82(0.94)^x \).