Question
Write an exponential function for the situation below and use your equation to answer the question.
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A population of 10 rabbits doubles in size every month. How many rabbits will there be after 8 months?
Use the Formula Editor x−−√ to type math or use the Image icon to upload a photo of your work.
A population of 10 rabbits doubles in size every month. How many rabbits will there be after 8 months?
Answers
Answered by
GPT-4o mini
To model the population of rabbits, we can use the exponential growth formula:
\[ P(t) = P_0 \cdot 2^{t} \]
Where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( t \) is the time in months,
- The base 2 indicates that the population doubles.
In this scenario:
- The initial population \( P_0 = 10 \) rabbits,
- The time \( t = 8 \) months.
Substituting the values into the equation:
\[ P(8) = 10 \cdot 2^{8} \]
Now we calculate \( 2^8 \):
\[ 2^8 = 256 \]
Now we can substitute \( 256 \) back into the equation:
\[ P(8) = 10 \cdot 256 \]
\[ P(8) = 2560 \]
Thus, after 8 months, the rabbit population will be **2560 rabbits**.
\[ P(t) = P_0 \cdot 2^{t} \]
Where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( t \) is the time in months,
- The base 2 indicates that the population doubles.
In this scenario:
- The initial population \( P_0 = 10 \) rabbits,
- The time \( t = 8 \) months.
Substituting the values into the equation:
\[ P(8) = 10 \cdot 2^{8} \]
Now we calculate \( 2^8 \):
\[ 2^8 = 256 \]
Now we can substitute \( 256 \) back into the equation:
\[ P(8) = 10 \cdot 256 \]
\[ P(8) = 2560 \]
Thus, after 8 months, the rabbit population will be **2560 rabbits**.