rabbit population doubles every month. If the population of rabbits is 32 at the end of the first month, how many rabbits will there be after eight months?(2 points)

Question

The initial number of rabbits is
. The common ratio is
. The number of rabbits after eight months is
.

Answer 1

To solve the problem, we need to consider how the rabbit population changes over time.

1. Initial Population: The population at the end of the first month is given as 32, which means there were 32 rabbits at the end of the first month. Let's denote this as the population at the start of the first month. Thus, the initial number of rabbits at the start of the first month is:

   • Initial Population: \( P_0 = 32 / 2 = 16 \) (since it doubles to 32 at the end of the month).

2. Common Ratio: Since the population doubles every month, the common ratio \( r \) is:

   • Common Ratio: \( r = 2 \).

3. Population After Eight Months: The formula for the population after \( n \) months is given by: \[ P(n) = P_0 \times r^n \] where \( P_0 \) is the initial population, \( r \) is the common ratio, and \( n \) is the number of months.

   For \( n = 8 \): \[ P(8) = 16 \times 2^8 \] Now we calculate \( 2^8 \): \[ 2^8 = 256 \]

   So, \[ P(8) = 16 \times 256 = 4096 \]

Now, summarizing the answers:

• The initial number of rabbits is 16.
• The common ratio is 2.
• The number of rabbits after eight months is 4096.