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Sagot :
Alright, let's evaluate the problem step-by-step and provide a detailed solution to understand what's true about the graph that represents the relationship between the temperature [tex]\( t \)[/tex] and the number of cricket chirps [tex]\( c \)[/tex] using the equation [tex]\( t = c + 40 \)[/tex].
1. Graph is Continuous:
- Consider the equation [tex]\( t = c + 40 \)[/tex]. Both [tex]\( t \)[/tex] (temperature) and [tex]\( c \)[/tex] (number of cricket chirps) can take any real number values, meaning that for any real value of [tex]\( c \)[/tex], there exists a corresponding real value of [tex]\( t \)[/tex].
- This implies that the relationship between [tex]\( t \)[/tex] and [tex]\( c \)[/tex] forms a continuous graph, since the values change smoothly without any jumps or gaps.
2. All values of [tex]\( t \)[/tex] must be positive:
- Let's analyze if [tex]\( t \)[/tex] always needs to be positive.
- If [tex]\( c \)[/tex] is a negative value and lower than -40, [tex]\( t \)[/tex] could be negative. For example, if [tex]\( c = -42 \)[/tex], then [tex]\( t = -42 + 40 = -2\)[/tex].
- This shows that [tex]\( t \)[/tex] is not restricted to only positive values and can indeed be negative if [tex]\( c \)[/tex] is a sufficiently negative value.
3. A viable solution is [tex]\((-2, 38)\)[/tex]:
- Plugging in [tex]\( c = -2 \)[/tex] into the equation: [tex]\( t = -2 + 40 = 38 \)[/tex].
- This solution satisfies the equation and is therefore viable.
4. A viable solution is [tex]\((0.5, 40.5)\)[/tex]:
- Plugging in [tex]\( c = 0.5 \)[/tex] into the equation: [tex]\( t = 0.5 + 40 = 40.5 \)[/tex].
- This solution also satisfies the equation and is viable.
5. A viable solution is [tex]\((10, 50)\)[/tex]:
- Plugging in [tex]\( c = 10 \)[/tex] into the equation: [tex]\( t = 10 + 40 = 50 \)[/tex].
- This solution satisfies the equation and is therefore viable.
Combining these observations, the true statements about the graph include:
1. The graph is continuous: This is true as explained above.
2. All values of [tex]\( t \)[/tex] must be positive: This is false because [tex]\( t \)[/tex] can indeed be negative.
3. A viable solution is [tex]\((-2, 38)\)[/tex]: This is true as verified by the calculation.
4. A viable solution is [tex]\((0.5, 40.5)\)[/tex]: This is true as verified by the calculation.
5. A viable solution is [tex]\((10, 50)\)[/tex]: This is true as verified by the calculation.
Therefore, the two correct options from the provided statements are:
- The graph is continuous.
- A viable solution is [tex]\((-2, 38)\)[/tex].
Additionally, viable solutions are [tex]\((0.5, 40.5)\)[/tex] and [tex]\((10, 50)\)[/tex], though only two selections were required.
1. Graph is Continuous:
- Consider the equation [tex]\( t = c + 40 \)[/tex]. Both [tex]\( t \)[/tex] (temperature) and [tex]\( c \)[/tex] (number of cricket chirps) can take any real number values, meaning that for any real value of [tex]\( c \)[/tex], there exists a corresponding real value of [tex]\( t \)[/tex].
- This implies that the relationship between [tex]\( t \)[/tex] and [tex]\( c \)[/tex] forms a continuous graph, since the values change smoothly without any jumps or gaps.
2. All values of [tex]\( t \)[/tex] must be positive:
- Let's analyze if [tex]\( t \)[/tex] always needs to be positive.
- If [tex]\( c \)[/tex] is a negative value and lower than -40, [tex]\( t \)[/tex] could be negative. For example, if [tex]\( c = -42 \)[/tex], then [tex]\( t = -42 + 40 = -2\)[/tex].
- This shows that [tex]\( t \)[/tex] is not restricted to only positive values and can indeed be negative if [tex]\( c \)[/tex] is a sufficiently negative value.
3. A viable solution is [tex]\((-2, 38)\)[/tex]:
- Plugging in [tex]\( c = -2 \)[/tex] into the equation: [tex]\( t = -2 + 40 = 38 \)[/tex].
- This solution satisfies the equation and is therefore viable.
4. A viable solution is [tex]\((0.5, 40.5)\)[/tex]:
- Plugging in [tex]\( c = 0.5 \)[/tex] into the equation: [tex]\( t = 0.5 + 40 = 40.5 \)[/tex].
- This solution also satisfies the equation and is viable.
5. A viable solution is [tex]\((10, 50)\)[/tex]:
- Plugging in [tex]\( c = 10 \)[/tex] into the equation: [tex]\( t = 10 + 40 = 50 \)[/tex].
- This solution satisfies the equation and is therefore viable.
Combining these observations, the true statements about the graph include:
1. The graph is continuous: This is true as explained above.
2. All values of [tex]\( t \)[/tex] must be positive: This is false because [tex]\( t \)[/tex] can indeed be negative.
3. A viable solution is [tex]\((-2, 38)\)[/tex]: This is true as verified by the calculation.
4. A viable solution is [tex]\((0.5, 40.5)\)[/tex]: This is true as verified by the calculation.
5. A viable solution is [tex]\((10, 50)\)[/tex]: This is true as verified by the calculation.
Therefore, the two correct options from the provided statements are:
- The graph is continuous.
- A viable solution is [tex]\((-2, 38)\)[/tex].
Additionally, viable solutions are [tex]\((0.5, 40.5)\)[/tex] and [tex]\((10, 50)\)[/tex], though only two selections were required.
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