To solve this problem, we need to understand the concept of inverse variation, specifically when a variable [tex]\( y \)[/tex] varies inversely as the square of another variable [tex]\( x \)[/tex].
1. Inverse Variation Relationship:
When [tex]\( y \)[/tex] varies inversely as the square of [tex]\( x \)[/tex], it means that [tex]\( y \)[/tex] is proportional to [tex]\( \frac{1}{x^2} \)[/tex]. We can express this relationship with a constant of proportionality [tex]\( k \)[/tex] as:
[tex]\[
y = \frac{k}{x^2}
\][/tex]
2. Finding the Constant [tex]\( k \)[/tex]:
To determine the constant [tex]\( k \)[/tex], we use the given values. We are told that when [tex]\( x = 3 \)[/tex], [tex]\( y = 100 \)[/tex].
Substituting these values into the equation, we get:
[tex]\[
100 = \frac{k}{3^2}
\][/tex]
3. Solving for [tex]\( k \)[/tex]:
Simplifying the equation:
[tex]\[
100 = \frac{k}{9}
\][/tex]
Multiplying both sides by 9 to solve for [tex]\( k \)[/tex]:
[tex]\[
k = 100 \times 9
\][/tex]
[tex]\[
k = 900
\][/tex]
4. Formulating the Equation:
Now that we have the constant [tex]\( k \)[/tex], we substitute it back into the original inverse variation equation:
[tex]\[
y = \frac{900}{x^2}
\][/tex]
So, the equation that connects [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is:
[tex]\[
y = \frac{900}{x^2}
\][/tex]
This equation describes how [tex]\( y \)[/tex] varies inversely as the square of [tex]\( x \)[/tex].
