Question

James Smith has decided to start a fish farm on his property. He initially stocked his pond with 100 fish. He then carefully monitored the growth of this pond. In the first month he measured, the population had grown to 110 fish. The second month he found there were 122 fish. The third month had 131 fish, and in the fourth month there were 146 fish. Identify what the independent and dependent variables represent and identify the domain (what are reasonable input values) and range (what are the expected output values) of this data. Let x be the domain variable and y be the range variable.

Answer 1

The independent variable x is time (in months), while the dependent variable y is the fish population. The domain for x is positive integers, and the range for y is
`\(y \geq 100\).`

The analysis of the data

|Variable | Description | Domain | Range|

|-------------|---------------------|-----------------|-----------|

| x | Month | 0, 1, 2, 3,.. | |

| y |Number of fish| 0, 1, 2, 3,...| |

In this scenario, the independent variable (domain variable) is the time elapsed, represented by x, while the dependent variable (range variable) is the fish population, represented by y.

- Independent Variable x: Time (in months). Time is independent as it is chosen or controlled by James, and it influences the fish population.

- Dependent Variable y: Fish Population. The number of fish depends on the time elapsed, and it is the variable being observed and measured.

- Domain (reasonable input values for x: x can take on values corresponding to the time elapsed. In this case, x is the set of positive integers representing months (1, 2, 3, ...).

- Range (expected output values for y): y represents the fish population, so the range is the set of positive integers greater than or equal to the initial population of 100
`(i.e., \(y \geq 100\))`.

In summary, the independent variable is time x, the dependent variable is fish population (y), the domain is positive integers representing months, and the range is positive integers greater than or equal to 100.