Answer :
Let’s analyze the information in the given question and go through each of the provided options step by step.
Given:
- The relationship between temperature [tex]\( t \)[/tex] in degrees Fahrenheit and the number of cricket chirps [tex]\( c \)[/tex] in 14 seconds is given by the equation [tex]\( t = c + 40 \)[/tex].
We need to evaluate the following statements about the graph representing this relationship:
1. The graph is continuous:
- Since [tex]\( t \)[/tex] is determined by adding a constant value to [tex]\( c \)[/tex], and both [tex]\( c \)[/tex] and [tex]\( t \)[/tex] can vary continuously (in real-world terms), the graph will be a straight line that extends infinitely in both directions without any breaks. Therefore, the graph is indeed continuous.
2. All values of [tex]\( t \)[/tex] must be positive:
- [tex]\( t = c + 40 \)[/tex] can potentially take any value depending on the value of [tex]\( c \)[/tex]. Even though it's uncommon to have negative chirps, theoretically [tex]\( c \)[/tex] could be negative, leading to a negative [tex]\( t \)[/tex]. Therefore, it is not necessarily true that all values of [tex]\( t \)[/tex] must be positive.
3. A viable solution is [tex]\( (-2, 38) \)[/tex]:
- This point implies that for [tex]\( c = -2 \)[/tex], [tex]\( t \)[/tex] would be 38. Plugging [tex]\( c = -2 \)[/tex] into the equation [tex]\( t = -2 + 40 = 38 \)[/tex]. Mathematically it holds true, but in practical terms, a negative chirp count is not logical. Hence, [tex]\( (-2, 38) \)[/tex] is not a viable solution in the context of counting chirps.
4. A viable solution is [tex]\( (0.5, 40.5) \)[/tex]:
- This point implies that for [tex]\( c = 0.5 \)[/tex], [tex]\( t \)[/tex] would be 40.5. Plugging [tex]\( c = 0.5 \)[/tex] into the equation [tex]\( t = 0.5 + 40 = 40.5 \)[/tex]. This is logically acceptable because fractional chirp counts could happen in cases of averaging over several measurements. So [tex]\( (0.5, 40.5) \)[/tex] is a viable solution.
5. A viable solution is [tex]\( (10, 50) \)[/tex]:
- This point implies that for [tex]\( c = 10 \)[/tex], [tex]\( t \)[/tex] would be 50. Plugging [tex]\( c = 10 \)[/tex] into the equation [tex]\( t = 10 + 40 = 50 \)[/tex]. This is straightforward and logically acceptable within the real-world context. So [tex]\( (10, 50) \)[/tex] is a viable solution.
Based on this detailed analysis, the two correct statements are:
- The graph is continuous.
- A viable solution is [tex]\( (0.5, 40.5) \)[/tex].
- A viable solution is [tex]\( (10, 50) \)[/tex].
Given:
- The relationship between temperature [tex]\( t \)[/tex] in degrees Fahrenheit and the number of cricket chirps [tex]\( c \)[/tex] in 14 seconds is given by the equation [tex]\( t = c + 40 \)[/tex].
We need to evaluate the following statements about the graph representing this relationship:
1. The graph is continuous:
- Since [tex]\( t \)[/tex] is determined by adding a constant value to [tex]\( c \)[/tex], and both [tex]\( c \)[/tex] and [tex]\( t \)[/tex] can vary continuously (in real-world terms), the graph will be a straight line that extends infinitely in both directions without any breaks. Therefore, the graph is indeed continuous.
2. All values of [tex]\( t \)[/tex] must be positive:
- [tex]\( t = c + 40 \)[/tex] can potentially take any value depending on the value of [tex]\( c \)[/tex]. Even though it's uncommon to have negative chirps, theoretically [tex]\( c \)[/tex] could be negative, leading to a negative [tex]\( t \)[/tex]. Therefore, it is not necessarily true that all values of [tex]\( t \)[/tex] must be positive.
3. A viable solution is [tex]\( (-2, 38) \)[/tex]:
- This point implies that for [tex]\( c = -2 \)[/tex], [tex]\( t \)[/tex] would be 38. Plugging [tex]\( c = -2 \)[/tex] into the equation [tex]\( t = -2 + 40 = 38 \)[/tex]. Mathematically it holds true, but in practical terms, a negative chirp count is not logical. Hence, [tex]\( (-2, 38) \)[/tex] is not a viable solution in the context of counting chirps.
4. A viable solution is [tex]\( (0.5, 40.5) \)[/tex]:
- This point implies that for [tex]\( c = 0.5 \)[/tex], [tex]\( t \)[/tex] would be 40.5. Plugging [tex]\( c = 0.5 \)[/tex] into the equation [tex]\( t = 0.5 + 40 = 40.5 \)[/tex]. This is logically acceptable because fractional chirp counts could happen in cases of averaging over several measurements. So [tex]\( (0.5, 40.5) \)[/tex] is a viable solution.
5. A viable solution is [tex]\( (10, 50) \)[/tex]:
- This point implies that for [tex]\( c = 10 \)[/tex], [tex]\( t \)[/tex] would be 50. Plugging [tex]\( c = 10 \)[/tex] into the equation [tex]\( t = 10 + 40 = 50 \)[/tex]. This is straightforward and logically acceptable within the real-world context. So [tex]\( (10, 50) \)[/tex] is a viable solution.
Based on this detailed analysis, the two correct statements are:
- The graph is continuous.
- A viable solution is [tex]\( (0.5, 40.5) \)[/tex].
- A viable solution is [tex]\( (10, 50) \)[/tex].
