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Sagot :
Let's analyze the information given and determine which table correctly represents the amount of popcorn produced by [tex]\( x \)[/tex] scoops of unpopped kernels.
Cory knows that 3 identical scoops of unpopped kernels produce 6 quarts of popcorn. This tells us that for every 3 scoops, 6 quarts of popcorn are produced.
To find the total amount of popcorn produced by [tex]\( x \)[/tex] scoops, we can set up a proportional relationship and derive a function [tex]\( f(x) \)[/tex].
Given:
[tex]\[ f(3) = 6 \][/tex]
This implies that each 3 scoops give 6 quarts, or more generally:
[tex]\[ f(x) = \frac{6}{3} \times x = 2x \][/tex]
Now, let’s verify each table by applying this function and checking if the values match.
### Table A:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline $x$ & 3 & 6 & 9 \\ \hline $f(x)$ & 6 & 12 & 18 \\ \hline \end{tabular} \][/tex]
Using [tex]\( f(x) = 2x \)[/tex]:
- For [tex]\( x = 3 \)[/tex]: [tex]\( f(3) = 2 \times 3 = 6 \)[/tex] ✅
- For [tex]\( x = 6 \)[/tex]: [tex]\( f(6) = 2 \times 6 = 12 \)[/tex] ✅
- For [tex]\( x = 9 \)[/tex]: [tex]\( f(9) = 2 \times 9 = 18 \)[/tex] ✅
### Table B:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline $x$ & 3 & 6 & 9 \\ \hline $f(x)$ & 1.5 & 3 & 4.5 \\ \hline \end{tabular} \][/tex]
Using [tex]\( f(x) = 2x \)[/tex]:
- For [tex]\( x = 3 \)[/tex]: [tex]\( f(3) = 2 \times 3 = 6 \)[/tex] ❌
- For [tex]\( x = 6 \)[/tex]: [tex]\( f(6) = 2 \times 6 = 12 \)[/tex] ❌
- For [tex]\( x = 9 \)[/tex]: [tex]\( f(9) = 2 \times 9 = 18 \)[/tex] ❌
None of these values match the function [tex]\( f(x) = 2x \)[/tex]. Therefore, Table B is incorrect.
### Table C:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline $x$ & 3 & 6 & 9 \\ \hline $f(x)$ & 6 & 12 & 24 \\ \hline \end{tabular} \][/tex]
Using [tex]\( f(x) = 2x \)[/tex]:
- For [tex]\( x = 3 \)[/tex]: [tex]\( f(3) = 2 \times 3 = 6 \)[/tex] ✅
- For [tex]\( x = 6 \)[/tex]: [tex]\( f(6) = 2 \times 6 = 12 \)[/tex] ✅
- For [tex]\( x = 9 \)[/tex]: [tex]\( f(9) = 2 \times 9 = 18 \)[/tex] ❌
The value for [tex]\( x = 9 \)[/tex] does not match. Therefore, Table C is incorrect.
### Table D:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline $x$ & 3 & 6 & 9 \\ \hline $f(x)$ & 1 & 3 & 9 \\ \hline \end{tabular} \][/tex]
Using [tex]\( f(x) = 2x \)[/tex]:
- For [tex]\( x = 3 \)[/tex]: [tex]\( f(3) = 2 \times 3 = 6 \)[/tex] ❌
- For [tex]\( x = 6 \)[/tex]: [tex]\( f(6) = 2 \times 6 = 12 \)[/tex] ❌
- For [tex]\( x = 9 \)[/tex]: [tex]\( f(9) = 2 \times 9 = 18 \)[/tex] ❌
None of these values match the function [tex]\( f(x) = 2x \)[/tex]. Therefore, Table D is incorrect.
### Conclusion:
After verifying against the given function [tex]\( f(x) = 2x \)[/tex], the table that correctly represents the relationship between [tex]\( x \)[/tex] scoops of unpopped kernels and the total amount of popcorn produced is:
[tex]\[ \boxed{\text{A}} \][/tex]
Cory knows that 3 identical scoops of unpopped kernels produce 6 quarts of popcorn. This tells us that for every 3 scoops, 6 quarts of popcorn are produced.
To find the total amount of popcorn produced by [tex]\( x \)[/tex] scoops, we can set up a proportional relationship and derive a function [tex]\( f(x) \)[/tex].
Given:
[tex]\[ f(3) = 6 \][/tex]
This implies that each 3 scoops give 6 quarts, or more generally:
[tex]\[ f(x) = \frac{6}{3} \times x = 2x \][/tex]
Now, let’s verify each table by applying this function and checking if the values match.
### Table A:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline $x$ & 3 & 6 & 9 \\ \hline $f(x)$ & 6 & 12 & 18 \\ \hline \end{tabular} \][/tex]
Using [tex]\( f(x) = 2x \)[/tex]:
- For [tex]\( x = 3 \)[/tex]: [tex]\( f(3) = 2 \times 3 = 6 \)[/tex] ✅
- For [tex]\( x = 6 \)[/tex]: [tex]\( f(6) = 2 \times 6 = 12 \)[/tex] ✅
- For [tex]\( x = 9 \)[/tex]: [tex]\( f(9) = 2 \times 9 = 18 \)[/tex] ✅
### Table B:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline $x$ & 3 & 6 & 9 \\ \hline $f(x)$ & 1.5 & 3 & 4.5 \\ \hline \end{tabular} \][/tex]
Using [tex]\( f(x) = 2x \)[/tex]:
- For [tex]\( x = 3 \)[/tex]: [tex]\( f(3) = 2 \times 3 = 6 \)[/tex] ❌
- For [tex]\( x = 6 \)[/tex]: [tex]\( f(6) = 2 \times 6 = 12 \)[/tex] ❌
- For [tex]\( x = 9 \)[/tex]: [tex]\( f(9) = 2 \times 9 = 18 \)[/tex] ❌
None of these values match the function [tex]\( f(x) = 2x \)[/tex]. Therefore, Table B is incorrect.
### Table C:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline $x$ & 3 & 6 & 9 \\ \hline $f(x)$ & 6 & 12 & 24 \\ \hline \end{tabular} \][/tex]
Using [tex]\( f(x) = 2x \)[/tex]:
- For [tex]\( x = 3 \)[/tex]: [tex]\( f(3) = 2 \times 3 = 6 \)[/tex] ✅
- For [tex]\( x = 6 \)[/tex]: [tex]\( f(6) = 2 \times 6 = 12 \)[/tex] ✅
- For [tex]\( x = 9 \)[/tex]: [tex]\( f(9) = 2 \times 9 = 18 \)[/tex] ❌
The value for [tex]\( x = 9 \)[/tex] does not match. Therefore, Table C is incorrect.
### Table D:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline $x$ & 3 & 6 & 9 \\ \hline $f(x)$ & 1 & 3 & 9 \\ \hline \end{tabular} \][/tex]
Using [tex]\( f(x) = 2x \)[/tex]:
- For [tex]\( x = 3 \)[/tex]: [tex]\( f(3) = 2 \times 3 = 6 \)[/tex] ❌
- For [tex]\( x = 6 \)[/tex]: [tex]\( f(6) = 2 \times 6 = 12 \)[/tex] ❌
- For [tex]\( x = 9 \)[/tex]: [tex]\( f(9) = 2 \times 9 = 18 \)[/tex] ❌
None of these values match the function [tex]\( f(x) = 2x \)[/tex]. Therefore, Table D is incorrect.
### Conclusion:
After verifying against the given function [tex]\( f(x) = 2x \)[/tex], the table that correctly represents the relationship between [tex]\( x \)[/tex] scoops of unpopped kernels and the total amount of popcorn produced is:
[tex]\[ \boxed{\text{A}} \][/tex]
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