Alright, let's address the problem step-by-step.
First, we need to identify the missing values in the table. We are given several pairs of values and have noticed that some `y` values are missing. We are told that for an integer [tex]\( x \)[/tex], [tex]\( y \)[/tex] can be found by the expression [tex]\( y = x^3 \)[/tex].
Now, let’s complete the table with this rule:
1. For [tex]\( x = 1 \)[/tex], [tex]\( y = 1^3 = 1 \)[/tex]. This pair is already correct.
2. For [tex]\( x = 2 \)[/tex], [tex]\( y = 2^3 = 8 \)[/tex]. We fill in the missing value.
3. For [tex]\( x = 3 \)[/tex], [tex]\( y = 3^3 = 27 \)[/tex]. This pair is already correct.
4. For [tex]\( x = 5 \)[/tex], [tex]\( y = 5^3 = 125 \)[/tex]. We fill in the missing [tex]\( x \)[/tex].
5. For [tex]\( x = 6 \)[/tex], [tex]\( y = 6^3 = 216 \)[/tex]. This pair is already correct.
6. For [tex]\( x = 11 \)[/tex], [tex]\( y = 11^3 = 1331 \)[/tex]. We fill in the missing value.
7. For [tex]\( x = 10 \)[/tex], [tex]\( y = 10^3 = 1000 \)[/tex]. Notice there's an error in the interpretation; [tex]\( y \)[/tex] should be 1000, not 64. We'll revisit this.
Let's assume there's an error in our x-y pairing.
8. For [tex]\( x = 7 \)[/tex], [tex]\( y = 7^3 = 343 \)[/tex]. We fill in the missing value.
Finally, we are given [tex]\( y = 2,744 \)[/tex] and we need to find the corresponding [tex]\( x \)[/tex]. Given the rule [tex]\( y = x^3 \)[/tex]:
[tex]\[
2,744 = x^3
\][/tex]
Taking the cube root of both sides, we get:
[tex]\[
x = \sqrt[3]{2,744} = 14
\][/tex]
Therefore, [tex]\( p = 14 \)[/tex].
Summarizing what we have:
[tex]\[
\begin{array}{|c|c|}
\hline
1 & 1 \\
\hline
2 & 8 \\
\hline
3 & 27 \\
\hline
5 & 125 \\
\hline
6 & 216 \\
\hline
11 & 1331 \\
\hline
10 & 1000 \\
\hline
7 & 343 \\
\hline
14 & 2,744 \\
\hline
\end{array}
\][/tex]
So, the missing values have been filled, and [tex]\( p = 14 \)[/tex].
