Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Get quick and reliable solutions to your questions from a community of experienced professionals on our platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
Let's analyze the data point by point using the function [tex]\( y = 5 \cdot 4^x \)[/tex] given by Julia.
1. Data Point: (0, 5)
- [tex]\( x = 0 \)[/tex]
- Using the function: [tex]\( y = 5 \cdot 4^0 = 5 \cdot 1 = 5 \)[/tex]
- The calculated [tex]\( y \)[/tex] matches the given [tex]\( y \)[/tex].
2. Data Point: (1, 20)
- [tex]\( x = 1 \)[/tex]
- Using the function: [tex]\( y = 5 \cdot 4^1 = 5 \cdot 4 = 20 \)[/tex]
- The calculated [tex]\( y \)[/tex] matches the given [tex]\( y \)[/tex].
3. Data Point: (2, 80)
- [tex]\( x = 2 \)[/tex]
- Using the function: [tex]\( y = 5 \cdot 4^2 = 5 \cdot 16 = 80 \)[/tex]
- The calculated [tex]\( y \)[/tex] matches the given [tex]\( y \)[/tex].
4. Data Point: (4, 320)
- [tex]\( x = 4 \)[/tex]
- Using the function: [tex]\( y = 5 \cdot 4^4 = 5 \cdot 256 = 1280 \)[/tex]
- The calculated [tex]\( y \)[/tex] does not match the given [tex]\( y \)[/tex] (given [tex]\( y = 320 \)[/tex]).
5. Data Point: (8, 640)
- [tex]\( x = 8 \)[/tex]
- Using the function: [tex]\( y = 5 \cdot 4^8 = 5 \cdot 65536 = 327680 \)[/tex]
- The calculated [tex]\( y \)[/tex] does not match the given [tex]\( y \)[/tex] (given [tex]\( y = 640 \)[/tex]).
### Conclusion
Based on the calculations, we observe that the given function [tex]\( y = 5 \cdot 4^x \)[/tex] correctly models the data points [tex]\( (0, 5) \)[/tex], [tex]\( (1, 20) \)[/tex], and [tex]\( (2, 80) \)[/tex], but it does not match the points [tex]\( (4, 320) \)[/tex] and [tex]\( (8, 640) \)[/tex].
Now we will determine which of the given statements is true:
1. Julia is correct because the distance starts at 5 feet and increases by a factor of 4.
- This statement is misleading because, although the distance starts at 5 feet and initially increases by a factor of 4, it does not increase consistently by a factor of 4 for all points.
2. Julia is correct because the function is true for [tex]\((0, 5)\)[/tex] and [tex]\((1, 20)\)[/tex].
- This statement is partially true but incomplete since the function is also true for [tex]\((2, 80)\)[/tex]. However, it fails for other points, so it is not completely true.
3. Julia is not correct because the function is not true for the point [tex]\((2, 80)\)[/tex].
- This statement is false because the function does correctly predict the point [tex]\((2, 80)\)[/tex].
4. Julia is not correct because the distance does not increase by a constant factor each minute.
- This statement is true. Although the function predicts an initial consistent multiplication factor, it does not hold for all data points provided in the table, indicating that Julia's model [tex]\( y = 5 \cdot 4^x \)[/tex] is not consistently accurate across all data points.
Thus, the correct statement is:
Julia is not correct because the distance does not increase by a constant factor each minute.
1. Data Point: (0, 5)
- [tex]\( x = 0 \)[/tex]
- Using the function: [tex]\( y = 5 \cdot 4^0 = 5 \cdot 1 = 5 \)[/tex]
- The calculated [tex]\( y \)[/tex] matches the given [tex]\( y \)[/tex].
2. Data Point: (1, 20)
- [tex]\( x = 1 \)[/tex]
- Using the function: [tex]\( y = 5 \cdot 4^1 = 5 \cdot 4 = 20 \)[/tex]
- The calculated [tex]\( y \)[/tex] matches the given [tex]\( y \)[/tex].
3. Data Point: (2, 80)
- [tex]\( x = 2 \)[/tex]
- Using the function: [tex]\( y = 5 \cdot 4^2 = 5 \cdot 16 = 80 \)[/tex]
- The calculated [tex]\( y \)[/tex] matches the given [tex]\( y \)[/tex].
4. Data Point: (4, 320)
- [tex]\( x = 4 \)[/tex]
- Using the function: [tex]\( y = 5 \cdot 4^4 = 5 \cdot 256 = 1280 \)[/tex]
- The calculated [tex]\( y \)[/tex] does not match the given [tex]\( y \)[/tex] (given [tex]\( y = 320 \)[/tex]).
5. Data Point: (8, 640)
- [tex]\( x = 8 \)[/tex]
- Using the function: [tex]\( y = 5 \cdot 4^8 = 5 \cdot 65536 = 327680 \)[/tex]
- The calculated [tex]\( y \)[/tex] does not match the given [tex]\( y \)[/tex] (given [tex]\( y = 640 \)[/tex]).
### Conclusion
Based on the calculations, we observe that the given function [tex]\( y = 5 \cdot 4^x \)[/tex] correctly models the data points [tex]\( (0, 5) \)[/tex], [tex]\( (1, 20) \)[/tex], and [tex]\( (2, 80) \)[/tex], but it does not match the points [tex]\( (4, 320) \)[/tex] and [tex]\( (8, 640) \)[/tex].
Now we will determine which of the given statements is true:
1. Julia is correct because the distance starts at 5 feet and increases by a factor of 4.
- This statement is misleading because, although the distance starts at 5 feet and initially increases by a factor of 4, it does not increase consistently by a factor of 4 for all points.
2. Julia is correct because the function is true for [tex]\((0, 5)\)[/tex] and [tex]\((1, 20)\)[/tex].
- This statement is partially true but incomplete since the function is also true for [tex]\((2, 80)\)[/tex]. However, it fails for other points, so it is not completely true.
3. Julia is not correct because the function is not true for the point [tex]\((2, 80)\)[/tex].
- This statement is false because the function does correctly predict the point [tex]\((2, 80)\)[/tex].
4. Julia is not correct because the distance does not increase by a constant factor each minute.
- This statement is true. Although the function predicts an initial consistent multiplication factor, it does not hold for all data points provided in the table, indicating that Julia's model [tex]\( y = 5 \cdot 4^x \)[/tex] is not consistently accurate across all data points.
Thus, the correct statement is:
Julia is not correct because the distance does not increase by a constant factor each minute.
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
