To describe the variation given by the equation [tex]\(\frac{y}{x^2} = 5\)[/tex], let's follow these steps:
1. Identify the Given Equation:
The equation provided is:
[tex]\[
\frac{y}{x^2} = 5
\][/tex]
2. Rearrange the Equation:
To understand how [tex]\(y\)[/tex] and [tex]\(x\)[/tex] are related, let's rearrange the equation to solve for [tex]\(y\)[/tex]:
[tex]\[
\frac{y}{x^2} = 5
\][/tex]
Multiply both sides of the equation by [tex]\(x^2\)[/tex]:
[tex]\[
y = 5x^2
\][/tex]
3. Understand the Relationship:
The rearranged equation shows that [tex]\(y\)[/tex] is equal to some constant (in this case, 5) multiplied by [tex]\(x\)[/tex] squared.
4. Determine the Type of Variation:
When we say that [tex]\(y\)[/tex] varies directly as another term, it means there is a constant [tex]\(k\)[/tex] such that [tex]\(y = k \times (\text{term})\)[/tex]. In this case:
[tex]\[
y = 5x^2
\][/tex]
Here, [tex]\(k\)[/tex], the constant of variation, is 5, and the term is [tex]\(x^2\)[/tex].
5. Conclude the Type of Variation:
Since [tex]\(y\)[/tex] is expressed as a constant times the square of [tex]\(x\)[/tex], we can conclude that [tex]\(y\)[/tex] varies directly as the square of [tex]\(x\)[/tex].
Therefore, the correct description of the variation given by the equation [tex]\(\frac{y}{x^2} = 5\)[/tex] is:
C. [tex]\(y\)[/tex] varies directly as the square of [tex]\(x\)[/tex].
