Let's analyze the given model [tex]\( t = c + 40 \)[/tex] and determine the correctness of each statement in the context provided.
### Viable Solutions:
We need to confirm if the given points are viable solutions by substituting [tex]\( c \)[/tex] into the equation [tex]\( t = c + 40 \)[/tex].
1. A viable solution is [tex]\((10, 50)\)[/tex]:
- For [tex]\( c = 10 \)[/tex]:
[tex]\[
t = 10 + 40 = 50
\][/tex]
- This point satisfies the equation, making [tex]\((10, 50)\)[/tex] a viable solution.
2. A viable solution is [tex]\((0.5, 40.5)\)[/tex]:
- For [tex]\( c = 0.5 \)[/tex]:
[tex]\[
t = 0.5 + 40 = 40.5
\][/tex]
- This point satisfies the equation, making [tex]\((0.5, 40.5)\)[/tex] a viable solution.
3. A viable solution is [tex]\((-2, 38)\)[/tex]:
- For [tex]\( c = -2 \)[/tex]:
[tex]\[
t = -2 + 40 = 38
\][/tex]
- Although the point satisfies the equation mathematically, [tex]\( c \)[/tex] representing the number of cricket chirps cannot be negative. Thus, [tex]\((-2, 38)\)[/tex] is not a viable solution in the real-world context.
### Graph Characteristics:
Next, we examine the properties of the graph representing the equation [tex]\( t = c + 40 \)[/tex]:
1. The graph is continuous:
- Both [tex]\( c \)[/tex] and [tex]\( t \)[/tex] are essentially continuous variables in real-world observations (even though [tex]\( c \)[/tex] would typically be an integer in counts), and therefore, the graph representing this equation would be continuous.
2. All values of [tex]\( t \)[/tex] must be positive:
- This is not necessarily true since if [tex]\( c \)[/tex] is negative (which theoretically could happen if considering the mathematical domain without physical constraints), [tex]\( t \)[/tex] could be less than 40, potentially negative. However, chirps, [tex]\( c \)[/tex], in real life should be non-negative, implying a minimum temperature of 40. Still, as per the strict mathematical interpretation, this statement is not universally true.
### Summary:
Based on the analysis, the two correct statements about the graph that represents this real-world scenario are:
1. The graph is continuous.
2. A viable solution is [tex]\((0.5, 40.5)\)[/tex].
3. A viable solution is [tex]\( (10, 50) \)[/tex].
Thus, the selected options are:
- The graph is continuous.
- A viable solution is [tex]\((0.5, 40.5)\)[/tex].
- A viable solution is [tex]\((10, 50)\)[/tex].
As we're asked to select two, the two viable and correct options to be highlighted would be:
- The graph is continuous.
- A viable solution is [tex]\((10, 50)\)[/tex].
Or alternately, noting the solution's nature, these combinations can work:
- The graph is continuous.
- A viable solution is [tex]\((0.5, 40.5)\)[/tex].
Either combination correctly identifies true aspects of the graph and the provided model.
