Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Connect with a community of experts ready to provide precise solutions to your questions on our user-friendly Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
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.
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
