To determine the domain of the function [tex]\( r(t) \)[/tex] which models the monthly rainfall in a city, let's analyze the context and characteristics of the function given:
1. Understanding the Context:
- The function [tex]\( r(t) \)[/tex] represents the monthly rainfall.
- The variable [tex]\( t \)[/tex] represents time in months from when the rainfall started being recorded.
- Rainfall occurs periodically with maximums and minimums at certain times of the year.
2. Characteristics of Trigonometric Functions:
- Trigonometric functions such as sine and cosine are periodic and their domains are usually all real numbers ([tex]\(-\infty, \infty\)[/tex]).
- These functions can model periodic events like monthly rainfall, which implies the phenomenon repeats over time.
3. Domain Analysis:
- Time [tex]\( t \)[/tex] represents the number of months elapsed, which means [tex]\( t \)[/tex] can be any real value within the context of the problem.
- Since we are discussing monthly rainfall and time progress, [tex]\( t \)[/tex] can be negative (representing a point before the reference time when recording started), zero, or positive.
4. Reviewing the Given Options:
- Option A [tex]\([0, 8]\)[/tex]: This represents a range of values, specifically from 0 to 8, which does not make sense for time as it would restrict the duration to just a fixed period of 8 months.
- Option B [tex]\((-\infty, \infty)\)[/tex]: This represents all real numbers, which means time can be any real value, fitting the context of a ongoing record of rainfall over months.
- Option C [tex]\([8, \infty]\)[/tex]: This suggests time starts at 8 months and only takes positive values afterward; it doesn't encompass negative or zero values.
- Option D [tex]\([0, \infty]\)[/tex]: This treats time as starting from 0 onwards; while it makes sense to consider current and future months, it negates the possibility of referencing past times ([tex]\( t \)[/tex] less than 0).
Given the above analysis, the domain that correctly represents the function [tex]\( r(t) \)[/tex] in the context of modeling monthly rainfall over any period of time is:
[tex]\[ B. (-\infty, \infty) \][/tex]
Hence, the domain of the function [tex]\( r(t) \)[/tex] is all real numbers.
