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Sagot :
To determine if the given data in the table can be modeled by the function [tex]\( y = 5(4)^x \)[/tex], we will verify the function at each time point and check the resulting distance. Let's go through the data step-by-step:
### Verifying the Function with Given Points
#### For Time = 0 minutes:
[tex]\[ y = 5 \cdot 4^0 = 5 \][/tex]
At [tex]\( x = 0 \)[/tex], the function gives us a distance of 5 feet, which matches the table.
#### For Time = 1 minute:
[tex]\[ y = 5 \cdot 4^1 = 20 \][/tex]
At [tex]\( x = 1 \)[/tex], the function gives us a distance of 20 feet, which matches the table.
#### For Time = 2 minutes:
[tex]\[ y = 5 \cdot 4^2 = 80 \][/tex]
At [tex]\( x = 2 \)[/tex], the function gives us a distance of 80 feet, which matches the table.
#### For Time = 4 minutes:
[tex]\[ y = 5 \cdot 4^4 = 5 \cdot 256 = 1280 \][/tex]
At [tex]\( x = 4 \)[/tex], the function gives us a distance of 1280 feet, which does not match the table value of 320 feet.
#### For Time = 8 minutes:
[tex]\[ y = 5 \cdot 4^8 = 5 \cdot 65536 = 327680 \][/tex]
At [tex]\( x = 8 \)[/tex], the function gives us a distance of 327680 feet, which is significantly larger than the table value of 640 feet.
Based on these calculations:
- The function [tex]\( y = 5(4)^x \)[/tex] accurately predicts the distances for [tex]\( x = 0, 1, \)[/tex] and [tex]\( 2 \)[/tex] but fails at [tex]\( x = 4 \)[/tex] and [tex]\( x = 8 \)[/tex].
### Evaluating the Statements
1. Julia is correct because the distance starts at 5 feet and increases by a factor of 4.
- This statement is ambiguous. While the distance does start at 5 feet and initially increases, it does not consistently increase by a factor of 4 each minute throughout the table.
2. Julia is correct because the function is true for [tex]\((0,5)\)[/tex] and [tex]\((1,20)\)[/tex].
- This statement is accurate, as the calculated distances for these points match the function.
3. Julia is not correct because the function is not true for the point [tex]\((2,80)\)[/tex].
- This statement is incorrect, as the calculated distance for [tex]\( x = 2 \)[/tex] matches 80 feet, which is in the table.
4. Julia is not correct because the distance does not increase by a constant factor each minute.
- This statement is somewhat misleading, but it is true upon checking the points [tex]\( x=4 \)[/tex] and [tex]\( x=8 \)[/tex].
### Final Conclusion:
- The true statements about Julia’s findings are:
- Statement 2: Julia is correct because the function is true for [tex]\((0,5)\)[/tex] and [tex]\((1,20)\)[/tex].
- Statement 4: Julia is not correct because the distance does not increase by a constant factor each minute.
Therefore, the answer is:
- Julia is correct because the function is true for [tex]\((0,5)\)[/tex] and [tex]\((1,20)\)[/tex].
- Julia is not correct because the distance does not increase by a constant factor each minute.
### Verifying the Function with Given Points
#### For Time = 0 minutes:
[tex]\[ y = 5 \cdot 4^0 = 5 \][/tex]
At [tex]\( x = 0 \)[/tex], the function gives us a distance of 5 feet, which matches the table.
#### For Time = 1 minute:
[tex]\[ y = 5 \cdot 4^1 = 20 \][/tex]
At [tex]\( x = 1 \)[/tex], the function gives us a distance of 20 feet, which matches the table.
#### For Time = 2 minutes:
[tex]\[ y = 5 \cdot 4^2 = 80 \][/tex]
At [tex]\( x = 2 \)[/tex], the function gives us a distance of 80 feet, which matches the table.
#### For Time = 4 minutes:
[tex]\[ y = 5 \cdot 4^4 = 5 \cdot 256 = 1280 \][/tex]
At [tex]\( x = 4 \)[/tex], the function gives us a distance of 1280 feet, which does not match the table value of 320 feet.
#### For Time = 8 minutes:
[tex]\[ y = 5 \cdot 4^8 = 5 \cdot 65536 = 327680 \][/tex]
At [tex]\( x = 8 \)[/tex], the function gives us a distance of 327680 feet, which is significantly larger than the table value of 640 feet.
Based on these calculations:
- The function [tex]\( y = 5(4)^x \)[/tex] accurately predicts the distances for [tex]\( x = 0, 1, \)[/tex] and [tex]\( 2 \)[/tex] but fails at [tex]\( x = 4 \)[/tex] and [tex]\( x = 8 \)[/tex].
### Evaluating the Statements
1. Julia is correct because the distance starts at 5 feet and increases by a factor of 4.
- This statement is ambiguous. While the distance does start at 5 feet and initially increases, it does not consistently increase by a factor of 4 each minute throughout the table.
2. Julia is correct because the function is true for [tex]\((0,5)\)[/tex] and [tex]\((1,20)\)[/tex].
- This statement is accurate, as the calculated distances for these points match the function.
3. Julia is not correct because the function is not true for the point [tex]\((2,80)\)[/tex].
- This statement is incorrect, as the calculated distance for [tex]\( x = 2 \)[/tex] matches 80 feet, which is in the table.
4. Julia is not correct because the distance does not increase by a constant factor each minute.
- This statement is somewhat misleading, but it is true upon checking the points [tex]\( x=4 \)[/tex] and [tex]\( x=8 \)[/tex].
### Final Conclusion:
- The true statements about Julia’s findings are:
- Statement 2: Julia is correct because the function is true for [tex]\((0,5)\)[/tex] and [tex]\((1,20)\)[/tex].
- Statement 4: Julia is not correct because the distance does not increase by a constant factor each minute.
Therefore, the answer is:
- Julia is correct because the function is true for [tex]\((0,5)\)[/tex] and [tex]\((1,20)\)[/tex].
- Julia is not correct because the distance does not increase by a constant factor each minute.
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