To find an equation for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] when [tex]\( y \)[/tex] is inversely proportional to the square of [tex]\( x \)[/tex], we can start by expressing the relationship mathematically:
[tex]\[ y \propto \frac{1}{x^2} \][/tex]
This can be rewritten as:
[tex]\[ y = \frac{k}{x^2} \][/tex]
where [tex]\( k \)[/tex] is a constant.
To determine the value of [tex]\( k \)[/tex], we use the given pairs of values for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the table.
From the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 2.25 \\
2 & 1 \\
3 & 0.25 \\
9 & 0.027777777777777776 \\
\hline
\end{array}
\][/tex]
We'll start with the first pair [tex]\((x, y) = (1, 2.25)\)[/tex]:
[tex]\[ y = \frac{k}{x^2} \][/tex]
By substituting [tex]\( x = 1 \)[/tex] and [tex]\( y = 2.25 \)[/tex] into the equation, we find [tex]\( k \)[/tex]:
[tex]\[ 2.25 = \frac{k}{1^2} \][/tex]
[tex]\[ k = 2.25 \][/tex]
Now that we have [tex]\( k = 2.25 \)[/tex], we can write the equation for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] as follows:
[tex]\[ y = \frac{2.25}{x^2} \][/tex]
Let's verify this equation with the other values in the table:
1. For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = \frac{2.25}{2^2} = \frac{2.25}{4} = 0.5625 \][/tex]
2. For [tex]\( x = 3 \)[/tex]:
[tex]\[ y = \frac{2.25}{3^2} = \frac{2.25}{9} = 0.25 \][/tex]
3. For [tex]\( x = 9 \)[/tex]:
[tex]\[ y = \frac{2.25}{9^2} = \frac{2.25}{81} = 0.027777777777777776 \][/tex]
These values confirm that our equation [tex]\( y = \frac{2.25}{x^2} \)[/tex] is correct for the given pairs of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
Therefore, the equation for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] is:
[tex]\[ y = \frac{2.25}{x^2} \][/tex]
