Multiplying Rabbits
Suppose that a pair of rabbits produces six offsrping: 3 males and 3 females. Assumes that no offspring die.
1. If each pair of rabbits breeds only once, how many offspring would be produced each year for 5 generations?
2. Construct a graph of your data. Plot time on the x-axis and population on the y-axis. What type of growth is the rabbit population going through after 5 years? (I just want to know how it would look like)
If there are 6 offspring, with 3 male and 3 female, then there are 3 pairs. If each pair produces 6 offspring then there will be 18. That then makes 9 pairs. That would be 54 rabbits or 27 pairs. If each of the 27 pairs produce 6 offspring there would be 162 offspring. This would leave 81 breeding pairs, which would make a total of 486 rabbits in the 5th generation.
1. To determine the number of offspring produced for each generation over 5 years, we can set up a table:
Generation | Number of pairs | Offspring per pair | Total offspring
------------|-----------------|----------------------|------------------
1 | 1 | 6 | 6
2 | 6 | 6 | 36
3 | 36 | 6 | 216
4 | 216 | 6 | 1,296
5 | 1,296 | 6 | 7,776
For the first year, there is only one pair of rabbits that produces 6 offspring. In the second year, each of the 6 pairs produced by the initial rabbit pair produces 6 offspring, resulting in a total of 36 offspring. This pattern continues for each subsequent generation.
To calculate the total offspring produced over 5 years, we can sum up the total offspring from each generation: 6 + 36 + 216 + 1,296 + 7,776 = 9,330 offspring.
2. The graph of the rabbit population over 5 years would show exponential growth. The x-axis represents time (in years), and the y-axis represents the population size. The graph would start with a small population size and then rapidly increase in a curved upward pattern as each generation produces more offspring.
To answer the first question, we need to understand the breeding pattern of rabbits. Rabbits breed in pairs, and each pair produces six offspring: three males and three females. Assuming no offspring die, we can determine the number of offspring produced for each generation.
1. First Generation: We start with a single pair of rabbits, which produces six offspring.
- Total Offspring: 6
2. Second Generation: The initial pair of rabbits matures and produces another six offspring, while each of their offspring from the first generation also matures and produces another six offspring.
- Initial Pair Offspring: 6
- Offspring from First Generation: (3 males + 3 females) * 6 = 36
- Total Offspring: 6 + 36 = 42
3. Third Generation: Similar to the second generation, both the initial pair and the offspring from the second generation reproduce.
- Initial Pair Offspring: 6
- Offspring from Second Generation: (6 pairs * 6 offspring) * 6 = 216
- Total Offspring: 6 + 216 = 222
4. Fourth Generation: Both the initial pair and the offspring from the third generation reproduce.
- Initial Pair Offspring: 6
- Offspring from Third Generation: (6 pairs * 6 offspring) * 6 = 216
- Total Offspring: 6 + 216 = 222
5. Fifth Generation: Similar to the previous generations, both the initial pair and the offspring from the fourth generation reproduce.
- Initial Pair Offspring: 6
- Offspring from Fourth Generation: (6 pairs * 6 offspring) * 6 = 216
- Total Offspring: 6 + 216 = 222
Therefore, for 5 generations, the number of offspring produced each year would be 222.
To construct the graph, we can plot time on the x-axis and the population on the y-axis. Assuming we start with one pair of rabbits in the first generation:
Time (generations) Population
0 1
1 6
2 42
3 222
4 222
5 222
The graph will show exponential growth for the first few generations, but it will level off once the number of offspring stabilizes.