To solve this problem, we can use the concept of exponential growth since the rabbit population doubles every month.
-
Initial number of rabbits: At the end of the first month, the population is 32. Therefore, the initial number of rabbits at the start of the first month (i.e., before any doubling) is half of 32.
\[ \text{Initial number of rabbits} = \frac{32}{2} = 16 \]
-
Common ratio: Since the population doubles every month, the common ratio \( r \) is 2.
\[ \text{Common ratio} = 2 \]
-
Number of rabbits after eight months: Given that the population doubles each month, we can use the formula for exponential growth:
\[ P(t) = P_0 \times r^t \]
where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the common ratio, and
- \( t \) is the number of time periods (months in this case).
For this problem:
- \( P_0 = 16 \) (initial population),
- \( r = 2 \) (common ratio),
- \( t = 8 \) (number of months).
Plugging these values into the formula:
\[ P(8) = 16 \times 2^8 \]
We first calculate \( 2^8 \):
\[ 2^8 = 256 \]
Now calculate \( P(8) \):
\[ P(8) = 16 \times 256 = 4096 \]
Let's summarize the results:
- The initial number of rabbits is 16.
- The common ratio is 2.
- The number of rabbits after eight months is 4096.
Thus:
- The initial number of rabbits is 16.
- The common ratio is 2.
- The number of rabbits after eight months is 4096.