Question

the population of a current species of rhinoceros is declining at a rate of 6% each year. There are currently only 82 rhinoceroses remaining. Create an exponential function to model the population decline. Which of the following options correctly models the decline? Option #1: f(x)=82(1.06)x Option #2: f(x)=82(1.6)x Option #3: f(x)=82(0.6)x Option #4: f(x)=82(0.94)x

Answer 1

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: \( f(x) = 82(0.94)^x \)**.