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 can use an exponential decay function of the form:

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

where:
- \( P_0 \) is the initial population, which is 82 in this case.
- \( r \) is the decay rate as a decimal. Since the population is declining at a rate of 6%, we have \( r = 0.06 \).
- \( x \) is the number of years.

Given that the population is decreasing, we should use \( (1 - r) \):

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

Now, let's examine the options given:

- Option #1: \( f(x)=82(1.06)^x \) (This indicates growth, not decay)
- Option #2: \( f(x)=82(1.6)^x \) (Incorrect due to incorrect growth)
- Option #3: \( f(x)=82(0.6)^x \) (This indicates a 40% decay, but we need 6% decay)
- Option #4: \( f(x)=82(0.94)^x \) (This is the correct model for a 6% decay)

Thus, the option that correctly models the problem is **Option #4: \( f(x)=82(0.94)^x \)**.