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?(1 point) Responses f(x)=82(0.94)x f of x is equal to 82 times 0 point 9 4 to the x th power f(x)=82(1.06)x f of x is equal to 82 times 1 point 0 6 to the x th power f(x)=82(0.06)x f of x is equal to 82 times 0 point 0 6 to the x th power f(x)=82(1.6)x
Answers
GPT-4o mini
To model the population decline of the rhinoceros, we can use an exponential decay function of the form:
\[ f(x) = P_0 \cdot (1 - r)^x \]
where:
- \( P_0 \) is the initial population,
- \( r \) is the rate of decline (as a decimal),
- \( x \) is the number of years.
Given:
- The initial population \( P_0 = 82 \),
- The rate of decline \( r = 6\% = 0.06 \).
Thus, \( 1 - r = 1 - 0.06 = 0.94 \).
So the exponential decay function modeling the population decline is:
\[ f(x) = 82 \cdot (0.94)^x \]
Therefore, the correct option that models the decline is:
**f(x) = 82(0.94)^x**.
\[ f(x) = P_0 \cdot (1 - r)^x \]
where:
- \( P_0 \) is the initial population,
- \( r \) is the rate of decline (as a decimal),
- \( x \) is the number of years.
Given:
- The initial population \( P_0 = 82 \),
- The rate of decline \( r = 6\% = 0.06 \).
Thus, \( 1 - r = 1 - 0.06 = 0.94 \).
So the exponential decay function modeling the population decline is:
\[ f(x) = 82 \cdot (0.94)^x \]
Therefore, the correct option that models the decline is:
**f(x) = 82(0.94)^x**.