A 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 can use the information given:

1. The rabbit population doubles every month.
2. The population at the end of the first month is 32 rabbits.

Let's break it down step by step:

• Initial Number of Rabbits: Since the population is 32 at the end of the first month, we need to find the initial population at the start. If it doubles every month, then at the start of the first month (before it doubled), the population would be half of 32:

  \[ \text{Initial population} = \frac{32}{2} = 16 \]

• Common Ratio: The common ratio is the factor by which the population multiplies each month. Since the population doubles, the common ratio is:

  \[ \text{Common ratio} = 2 \]

• Number of Rabbits After Eight Months: To find the rabbit population after 8 months, we can use the formula for exponential growth based on the initial population.

  \[ \text{Population after } n \text{ months} = \text{Initial population} \times (\text{Common ratio})^n \]

  Plugging in the values:

  \[ n = 8 \text{ months} \] \[ \text{Population after 8 months} = 16 \times 2^8 \]

  Now calculating \(2^8\):

  \[ 2^8 = 256 \]

  Therefore:

  \[ \text{Population after 8 months} = 16 \times 256 = 4096 \]

To summarize:

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