To solve this problem, we start with the information that [tex]\( y \)[/tex] is directly proportional to the square of [tex]\( x \)[/tex]. This relationship can be expressed mathematically as:
[tex]\[ y = kx^2 \][/tex]
where [tex]\( k \)[/tex] is the proportionality constant.
We are given that [tex]\( y = 8 \)[/tex] when [tex]\( x = 4 \)[/tex]. Substituting these values into the equation, we can solve for [tex]\( k \)[/tex]:
[tex]\[ 8 = k \cdot 4^2 \][/tex]
[tex]\[ 8 = k \cdot 16 \][/tex]
[tex]\[ k = \frac{8}{16} \][/tex]
[tex]\[ k = \frac{1}{2} \][/tex]
Now that we have determined the value of the constant [tex]\( k \)[/tex], we can use it to find the value of [tex]\( y \)[/tex] when [tex]\( x = 3 \)[/tex]:
[tex]\[ y = \frac{1}{2} \cdot (3)^2 \][/tex]
[tex]\[ y = \frac{1}{2} \cdot 9 \][/tex]
[tex]\[ y = \frac{9}{2} \][/tex]
[tex]\[ y = 4.5 \][/tex]
Therefore, when [tex]\( x = 3 \)[/tex], the value of [tex]\( y \)[/tex] is [tex]\( 4.5 \)[/tex].
