Certainly! Let's analyze the given relationships step-by-step.
1. The Equation: The equation provided is [tex]\( t = c + 40 \)[/tex], where:
- [tex]\( t \)[/tex] is the temperature in degrees Fahrenheit.
- [tex]\( c \)[/tex] is the number of cricket chirps in 14 seconds.
- This equation tells us that for any given number of cricket chirps [tex]\( c \)[/tex], the temperature [tex]\( t \)[/tex] is found by adding 40 to the number of chirps.
2. Graph Continuity:
- In this context, the number of chirps [tex]\( c \)[/tex] could theoretically take any value, including fractional (e.g., [tex]\( 0.5 \)[/tex]) or negative values (e.g., [tex]\( -2 \)[/tex]), even though negative chirp counts may not make practical sense.
- Because there is no restriction on the potential values of [tex]\( c \)[/tex] within the modeled relationship, the graph of the function [tex]\( t = c + 40 \)[/tex] is continuous.
3. Positivity of [tex]\( t \)[/tex]:
- Not all values of [tex]\( t \)[/tex] are necessarily positive. If [tex]\( c \)[/tex] were a sufficiently large negative number, [tex]\( c + 40 \)[/tex] could result in a negative value for [tex]\( t \)[/tex]. So, it is not required that all values of [tex]\( t \)[/tex] be positive.
4. Viable Solutions:
- To verify viable solutions, we need to substitute each pair [tex]\((c, t)\)[/tex] into the equation [tex]\( t = c + 40 \)[/tex] and check if it holds true:
- For [tex]\((-2, 38)\)[/tex]:
[tex]\[
38 = -2 + 40 \quad \text{(True)}
\][/tex]
- For [tex]\((0.5, 40.5)\)[/tex]:
[tex]\[
40.5 = 0.5 + 40 \quad \text{(True)}
\][/tex]
- For [tex]\((10, 50)\)[/tex]:
[tex]\[
50 = 10 + 40 \quad \text{(True)}
\][/tex]
Therefore:
- The graph is continuous.
- Not all values of [tex]\( t \)[/tex] must be positive.
- [tex]\((-2, 38)\)[/tex] is a viable solution.
- [tex]\((0.5, 40.5)\)[/tex] is a viable solution.
- [tex]\((10, 50)\)[/tex] is a viable solution.
Based on these deductions, the correct options you should select are:
1. The graph is continuous.
2. A viable solution is [tex]\((0.5, 40.5)\)[/tex].
These reflect the true characteristics of the graph and specific viable points on it.
