To determine the number of rabbits after eight months given that the population doubles every month and is 32 at the end of the first month, we can use the following information:
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Initial Number of Rabbits (after the first month): At the end of the first month, the population is 32. Therefore, the initial number (at the beginning of the first month) would be half of that: \[ 32 / 2 = 16 \]
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Common Ratio: Since the population doubles every month, the common ratio \( r \) is 2.
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Number of Rabbits after Eight Months: The rabbit population can be calculated using the formula for exponential growth: \[ P(t) = P_0 \cdot r^n \] where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, \( r \) is the common ratio, and \( n \) is the number of months.
Here:
- \( P_0 = 16 \)
- \( r = 2 \)
- \( n = 8 \)
Plugging the values into the formula: \[ P(8) = 16 \cdot 2^8 \]
First, calculate \( 2^8 \): \[ 2^8 = 256 \]
Then, calculate \( P(8) \): \[ P(8) = 16 \cdot 256 = 4096 \]
So, the solutions are:
- The initial number of rabbits is 16.
- The common ratio is 2.
- The number of rabbits after eight months is 4096.