Question

Ant Farming  You have an ant farm with 34 ants. The population of ants in your farm will double every 3 months. The table shows the population growth of the ants over nine months. Decide whether the table represents a linear function or a nonlinear function. After one​ year, how many ants will there be in the ant​ farm?
Population of Ants
Number of Months
0
3
6
9
Population
34
68
136
272
Question content area bottom
Part 1
The table represents a
▼
nonlinear
linear
function.

Answer 1

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.