To solve this problem, we follow a methodical approach based on the given information that [tex]\( y \)[/tex] varies directly as [tex]\( x^2 \)[/tex].
1. Identify the relationship:
Since [tex]\( y \)[/tex] varies directly as [tex]\( x^2 \)[/tex], we can express this relationship in the form of an equation:
[tex]\[
y = k x^2
\][/tex]
where [tex]\( k \)[/tex] is a constant.
2. Determine the constant [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]\[
12 = k (2^2)
\][/tex]
Simplify the equation:
[tex]\[
12 = k \cdot 4
\][/tex]
Divide both sides by 4 to isolate [tex]\( k \)[/tex]:
[tex]\[
k = \frac{12}{4}
\][/tex]
[tex]\[
k = 3
\][/tex]
3. Use the constant to find the new value of [tex]\( y \)[/tex]:
We are asked to find [tex]\( y \)[/tex] when [tex]\( x = 5 \)[/tex]. Substitute [tex]\( x = 5 \)[/tex] and [tex]\( k = 3 \)[/tex] into the equation [tex]\( y = k x^2 \)[/tex]:
[tex]\[
y = 3 (5^2)
\][/tex]
Simplify the equation:
[tex]\[
y = 3 \cdot 25
\][/tex]
[tex]\[
y = 75
\][/tex]
Thus, when [tex]\( x = 5 \)[/tex], the value of [tex]\( y \)[/tex] is 75.
