the population of a current species of rhinoceros is declining at a rate of 6% each year. there are currently only 82 rhinoceros remaining. create an exponential function to model the population decline. which of the following options correctly models the decline.

Question

a. f (x) = 82(1.06)^x
b. f (x) = 82(1.6)^x
c. f (x) = 82(0.6)^x
d. f (x) = 82(0.94)^x

Answer 1

To model the declining population of the rhinoceros species, we can use an exponential decay function. When a population declines by a percentage each year, we can represent this decline mathematically.

1. The initial population (at year \(x = 0\)) is 82 rhinoceroses.
2. The population declines at a rate of 6% each year. This means that each year, 94% of the population remains (since 100% - 6% = 94%).

Thus, we can model the remaining population after \(x\) years using the equation:

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

Where:

• \(P_0\) is the initial population (82 rhinoceroses),
• \(r\) is the rate of decline (0.06 for 6%),
• \(1 - r\) is the proportion that remains each year (0.94).

Therefore, the function becomes:

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

This corresponds to option d.

So, the correct choice is:

d. f(x) = 82(0.94)^x