Question: Select the correct answer.
A trigonometric function models the monthly rainfall of a city, which reaches a maximum monthly rainfall of $8$ inches in the spring and fall and a minimum monthly rainfall of $0$ inches in the summer and winter.
If $r(t)$ models the monthly rainfall $t$ months from when the monthly rainfall started being recorded for the city, what is the domain of function $r$?
1. **State the problem:** We need to find the domain of the function $r(t)$ that models monthly rainfall over time.
2. **Understand the context:** The function $r(t)$ represents rainfall $t$ months after recording started. Since $t$ represents time in months from the start, it cannot be negative.
3. **Domain of time-based functions:** Time usually starts at $0$ and continues indefinitely, so the domain is all $t$ such that $t \geq 0$.
4. **Check the options:**
- $[0,8]$: This restricts $t$ to between $0$ and $8$ months, which is too limited.
- $(-\infty, \infty)$: This allows negative time, which is not possible.
- $[8, \infty)$: This starts at $8$ months, ignoring the first $8$ months.
- $[0, \infty)$: This includes all months from the start onward, which fits the problem.
5. **Conclusion:** The domain of $r$ is $[0, \infty)$ because time starts at $0$ and continues indefinitely.
**Final answer:** $[0, \infty)$