To determine whether the table represents a linear function or a nonlinear function, we can analyze the population growth.
A linear function has a constant rate of change, meaning that the differences between the successive populations would be the same each time. In contrast, a nonlinear function has a variable rate of change.
From the data provided:
- At 0 months: 34 ants
- At 3 months: 68 ants
- At 6 months: 136 ants
- At 9 months: 272 ants
Let's calculate the changes in population:
- From 0 to 3 months: \(68 - 34 = 34\) ants
- From 3 to 6 months: \(136 - 68 = 68\) ants
- From 6 to 9 months: \(272 - 136 = 136\) ants
The differences between the populations are 34, 68, and 136, which are not constant; they are actually doubling each time (34, 68 is double of 34, and 136 is double of 68). Therefore, the function is nonlinear.
Conclusion:
The table represents a nonlinear function.
To find the population after one year (12 months):
Since the population doubles every 3 months, we can calculate how many times it will double in one year (12 months):
\(\text{Number of doublings in 12 months} = \frac{12 \text{ months}}{3 \text{ months/doubling}} = 4\) doublings
Starting with 34 ants, we can calculate the population after 4 doublings:
\[ \text{Population after 12 months} = 34 \times (2^4) = 34 \times 16 = 544 \]
Final Answer:
There will be 544 ants in the ant farm after one year.
2 answers