To determine the values for [tex]\(y\)[/tex] corresponding to the given [tex]\(x\)[/tex] values, we need to establish a relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. Let's start by analyzing the information we have.
Given values:
[tex]\[
\begin{tabular}{|c|c|}
\hline $x$ & $y$ \\
\hline 1 & 2 \\
\hline 10 & $a$ \\
\hline 100 & $b$ \\
\hline 10,000 & $c$ \\
\hline 100,000 & $d$ \\
\hline 1,000,000 & $e$ \\
\hline
\end{tabular}
\][/tex]
From the table, when [tex]\(x = 1\)[/tex], we have [tex]\( y = 2 \)[/tex]. This suggests [tex]\( y \)[/tex] might be a function of [tex]\( x \)[/tex]. A common and simple relationship we can try is a power or an exponential relationship.
Let's assume [tex]\( y \)[/tex] is related to [tex]\( x \)[/tex] by a simple power relationship of the form:
[tex]\[ y = k \cdot x^n \][/tex]
Given [tex]\( x = 1 \)[/tex], [tex]\( y = 2 \)[/tex]:
[tex]\[ 2 = k \cdot 1^n \][/tex]
[tex]\[ k = 2 \][/tex]
So our equation becomes:
[tex]\[ y = 2 \cdot x^n \][/tex]
Next, we need to find the value of [tex]\( n \)[/tex]. We do not have direct information to determine [tex]\( n \)[/tex] from the other values, but we need to assume it to proceed further. Let's assume [tex]\( n = 1 \)[/tex] as it is a simple and commonly seen linear relationship.
Using [tex]\( n = 1 \)[/tex]:
[tex]\[ y = 2 \cdot x \][/tex]
Now let's calculate the values:
1. For [tex]\( x = 10 \)[/tex]:
[tex]\[ y = 2 \cdot 10 = 20 \][/tex]
Thus, [tex]\( a = 20 \)[/tex].
2. For [tex]\( x = 100 \)[/tex]:
[tex]\[ y = 2 \cdot 100 = 200 \][/tex]
Thus, [tex]\( b = 200 \)[/tex].
3. For [tex]\( x = 10,000 \)[/tex]:
[tex]\[ y = 2 \cdot 10,000 = 20,000 \][/tex]
Thus, [tex]\( c = 20,000 \)[/tex].
4. For [tex]\( x = 100,000 \)[/tex]:
[tex]\[ y = 2 \cdot 100,000 = 200,000 \][/tex]
Thus, [tex]\( d = 200,000 \)[/tex].
5. For [tex]\( x = 1,000,000 \)[/tex]:
[tex]\[ y = 2 \cdot 1,000,000 = 2,000,000 \][/tex]
Thus, [tex]\( e = 2,000,000 \)[/tex].
Summarizing, the completed table is:
[tex]\[
\begin{tabular}{|c|c|}
\hline $x$ & $y$ \\
\hline 1 & 2 \\
\hline 10 & 20 \\
\hline 100 & 200 \\
\hline 10,000 & 20,000 \\
\hline 100,000 & 200,000 \\
\hline 1,000,000 & 2,000,000 \\
\hline
\end{tabular}
\][/tex]
