To solve for [tex]\( k \)[/tex] and [tex]\( n \)[/tex] given the equation [tex]\( z = \frac{k}{x^n} \)[/tex], let's use the data points provided:
\begin{itemize}
\item [tex]\( x = 1 \)[/tex], [tex]\( z = 100 \)[/tex]
\item [tex]\( x = 2 \)[/tex], [tex]\( z = 12 \frac{1}{2} = 12.5 \)[/tex]
\item [tex]\( x = 4 \)[/tex]
\item [tex]\( z = \frac{1}{10} = 0.1 \)[/tex]
\end{itemize}
### Step 1: Find [tex]\( k \)[/tex] using the first data point.
From the first data point [tex]\( (x = 1, z = 100) \)[/tex]:
[tex]\[
z = \frac{k}{x^n} \implies 100 = \frac{k}{1^n}
\][/tex]
Since [tex]\( 1^n = 1 \)[/tex]:
[tex]\[
100 = \frac{k}{1} \implies k = 100
\][/tex]
### Step 2: Find [tex]\( n \)[/tex] using the second data point.
From the second data point [tex]\( (x = 2, z = 12.5) \)[/tex]:
[tex]\[
z = \frac{k}{x^n} \implies 12.5 = \frac{100}{2^n}
\][/tex]
Solving for [tex]\( 2^n \)[/tex]:
[tex]\[
12.5 = \frac{100}{2^n} \implies 2^n = \frac{100}{12.5} \implies 2^n = 8 \implies n = 3
\][/tex]
So, we have:
[tex]\[
k = 100 \quad \text{and} \quad n = 3
\][/tex]
### Step 3: Complete the table.
#### Finding [tex]\( z \)[/tex] for [tex]\( x = 4 \)[/tex]:
[tex]\[
z = \frac{k}{x^n} = \frac{100}{4^3} = \frac{100}{64} \approx 1.56
\][/tex]
#### Finding [tex]\( x \)[/tex] for [tex]\( z = 0.1 \)[/tex]:
[tex]\[
0.1 = \frac{100}{x^3} \implies x^3 = \frac{100}{0.1} \implies x^3 = 1000 \implies x = \sqrt[3]{1000} = 10
\][/tex]
### Final Table:
[tex]\[
\begin{tabular}{c||c|c|c|c}
$x$ & 1 & 2 & 4 & 10 \\
\hline
$z$ & 100 & 12.5 & 1.56 & 0.1 \\
\end{tabular}
\][/tex]
Thus, we have [tex]\( k = 100 \)[/tex], [tex]\( n = 3 \)[/tex], and the completed table is:
[tex]\[
\begin{tabular}{c||c|c|c|c}
$x$ & 1 & 2 & 4 & 10 \\
\hline
$z$ & 100 & 12.5 & 1.56 & 0.1 \\
\end{tabular}
\][/tex]
