Question
the population of a current species of rhinoceros is declining at a rate of 6% each year. they are currently only 82 rhinoceroses reamining. create an expoential 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
the option that correctly models the problem is option # ____
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
the option that correctly models the problem is option # ____
Answers
GPT-4o mini
To model the decline of the rhinoceros population, we need to recognize that a declining population at a rate of 6% means that each year, 94% of the population remains (100% - 6% = 94%).
Therefore, the population can be modeled with an exponential decay function in the following form:
\[ f(x) = P_0 \cdot (1 - r)^x \]
where:
- \( P_0 \) is the initial population,
- \( r \) is the rate of decline (in decimal form),
- \( x \) is the number of years.
Given that:
- \( P_0 = 82 \)
- \( r = 0.06 \)
The correct model becomes:
\[ f(x) = 82 \cdot (1 - 0.06)^x \]
\[ f(x) = 82 \cdot (0.94)^x \]
Comparing this with the options you provided:
- Option #1: \( f(x)=82(1.06)^x \) (incorrect; this represents growth)
- Option #2: \( f(x)=82(1.6)^x \) (incorrect; this does not relate to the decline)
- Option #3: \( f(x)=82(0.6)^x \) (incorrect; this represents a decline of 40%)
- Option #4: \( f(x)=82(0.94)^x \) (correct)
Thus, the option that correctly models the population decline is option # **4**.
Therefore, the population can be modeled with an exponential decay function in the following form:
\[ f(x) = P_0 \cdot (1 - r)^x \]
where:
- \( P_0 \) is the initial population,
- \( r \) is the rate of decline (in decimal form),
- \( x \) is the number of years.
Given that:
- \( P_0 = 82 \)
- \( r = 0.06 \)
The correct model becomes:
\[ f(x) = 82 \cdot (1 - 0.06)^x \]
\[ f(x) = 82 \cdot (0.94)^x \]
Comparing this with the options you provided:
- Option #1: \( f(x)=82(1.06)^x \) (incorrect; this represents growth)
- Option #2: \( f(x)=82(1.6)^x \) (incorrect; this does not relate to the decline)
- Option #3: \( f(x)=82(0.6)^x \) (incorrect; this represents a decline of 40%)
- Option #4: \( f(x)=82(0.94)^x \) (correct)
Thus, the option that correctly models the population decline is option # **4**.