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


(1 point)
The option that correctly models the problem is Option #
.

Answer 1

To model the population decline of the rhinoceroses, we need to use an exponential decay function. The general form of the function to model decline is:

\[ f(x) = P_0 \cdot (1 - r)^x \]

where:
- \( P_0 \) is the initial population,
- \( r \) is the rate of decline (as a decimal),
- \( x \) is the number of years,
- \( (1 - r) \) represents the remaining population after each year.

Given the following:
- The current population \( P_0 = 82 \).
- The decline rate \( r = 0.06 \) (6%).

Now, we substitute these values into the formula:

\[ f(x) = 82 \cdot (1 - 0.06)^x = 82 \cdot (0.94)^x \]

Now, let's check the options given:

1. **Option #1**: \( f(x) = 82(1.06)^x \) (Incorrect, this is growth)
2. **Option #2**: \( f(x) = 82(1.6)^x \) (Incorrect, this is also growth)
3. **Option #3**: \( f(x) = 82(0.6)^x \) (Incorrect, the decline rate should be 0.94)
4. **Option #4**: \( f(x) = 82(0.94)^x \) (Correct)

Thus, the option that correctly models the population decline is **Option #4**.