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 decline of a population that is decreasing at a rate of 6% per year, we can use the formula:

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

where:
- \( P_0 \) is the initial population,
- \( r \) is the rate of decline (as a decimal),
- \( x \) is the number of years, and
- \( f(x) \) is the population after \( x \) years.

In this case:
- The initial population \( P_0 = 82 \),
- The rate of decline \( r = 0.06 \).

Therefore, the formula becomes:

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

Now let's evaluate the options:
- Option #1: \( f(x) = 82(1.06)^x \) (growing population, incorrect)
- Option #2: \( f(x) = 82(1.6)^x \) (growing population, incorrect)
- Option #3: \( f(x) = 82(0.6)^x \) (incorrect rate of decline)
- Option #4: \( f(x) = 82(0.94)^x \) (correct)

Thus, the option that correctly models the problem is Option #4.