Sure, let's find [tex]\( y \)[/tex] when [tex]\( x = 5 \)[/tex] given the relationship between [tex]\( y \)[/tex] and [tex]\( x^2 \)[/tex]. Here is the step-by-step process to solve this:
1. Understand the relationship and the given values:
We know that [tex]\( y \)[/tex] varies directly as [tex]\( x^2 \)[/tex]. Mathematically, this can be written as:
[tex]\[
y = k \cdot x^2
\][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality.
2. Use the given values to find [tex]\( k \)[/tex]:
We are given that [tex]\( y = 12 \)[/tex] when [tex]\( x = 2 \)[/tex]. Substitute these values into the equation to find [tex]\( k \)[/tex]:
[tex]\[
12 = k \cdot 2^2
\][/tex]
[tex]\[
12 = 4k
\][/tex]
Solving for [tex]\( k \)[/tex], we divide both sides by 4:
[tex]\[
k = \frac{12}{4} = 3
\][/tex]
3. Use the value of [tex]\( k \)[/tex] to find [tex]\( y \)[/tex] when [tex]\( x = 5 \)[/tex]:
Now that we have [tex]\( k = 3 \)[/tex], we can substitute [tex]\( x = 5 \)[/tex] into the equation [tex]\( y = k \cdot x^2 \)[/tex] to find [tex]\( y \)[/tex]:
[tex]\[
y = 3 \cdot 5^2
\][/tex]
[tex]\[
y = 3 \cdot 25
\][/tex]
[tex]\[
y = 75
\][/tex]
So, when [tex]\( x = 5 \)[/tex], [tex]\( y \)[/tex] is 75.
