Let's determine the type of variation modeled in the table with [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values.
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
x & 3 & 9 & 36 \\
\hline
y & 48 & 16 & 4 \\
\hline
\end{array}
\][/tex]
### Step 1: Calculate the Products of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] for Each Pair
First, we multiply each [tex]\( x \)[/tex] value by its corresponding [tex]\( y \)[/tex] value:
For [tex]\( x = 3 \)[/tex] and [tex]\( y = 48 \)[/tex],
[tex]\[ 3 \times 48 = 144 \][/tex]
For [tex]\( x = 9 \)[/tex] and [tex]\( y = 16 \)[/tex],
[tex]\[ 9 \times 16 = 144 \][/tex]
For [tex]\( x = 36 \)[/tex] and [tex]\( y = 4 \)[/tex],
[tex]\[ 36 \times 4 = 144 \][/tex]
### Step 2: Analyze the Products
We notice that the product [tex]\( x \times y \)[/tex] is the same for all pairs:
[tex]\[ 144, 144, 144 \][/tex]
Since the product [tex]\( x \times y \)[/tex] is constant (144 in all cases), the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is indicative of an inverse variation.
### Step 3: Confirm the Type of Variation
In an inverse variation, the product of the two variables remains constant. Mathematically, this can be written as:
[tex]\[ x \times y = k \][/tex]
where [tex]\( k \)[/tex] is a constant.
Given that the product is always 144, it confirms that the variation type is indeed an inverse variation.
### Conclusion
We can conclude that the type of variation modeled in the table is an inverse variation.
Answer: inverse
