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?

Question

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 note that the population is decreasing at a rate of 6% each year.

This means that each year, 94% of the population remains (since 100% - 6% = 94%). Therefore, we can express the remaining population as a fraction of the previous population:

• The exponential decay function can be written as: \[ f(x) = P_0 \cdot (1 - r)^x \] Where:
• \( P_0 \) is the initial population,
• \( r \) is the rate of decline,
• \( x \) is the number of years.

In this case:

• \( P_0 = 82 \)
• \( r = 0.06 \)

Thus, the formula becomes: \[ f(x) = 82 \cdot (1 - 0.06)^x \] \[ f(x) = 82 \cdot (0.94)^x \]

Among the provided options:

• Option #4: \( f(x) = 82(0.94)^x \) is the correct model for this population decline.

So, the correct answer is Option #4.