To solve this problem, we need to understand joint variation. Since [tex]\( g \)[/tex] varies jointly with [tex]\( h \)[/tex] and [tex]\( j \)[/tex], we can write the relationship as:
[tex]\[ g = k \cdot h \cdot j \][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality.
1. Find the constant [tex]\( k \)[/tex]:
We know that when [tex]\( h = \frac{1}{2} \)[/tex] and [tex]\( j = \frac{1}{3} \)[/tex], [tex]\( g = 4 \)[/tex]. Plugging these values into the equation gives us:
[tex]\[ 4 = k \cdot \frac{1}{2} \cdot \frac{1}{3} \][/tex]
Simplify the right side:
[tex]\[ 4 = k \cdot \frac{1}{6} \][/tex]
To isolate [tex]\( k \)[/tex], multiply both sides by 6:
[tex]\[ k = 4 \times 6 \][/tex]
[tex]\[ k = 24 \][/tex]
2. Use the constant [tex]\( k \)[/tex] to find the new [tex]\( g \)[/tex]:
Now, we need to find [tex]\( g \)[/tex] when [tex]\( h = 2 \)[/tex] and [tex]\( j = 3 \)[/tex]. Using the joint variation equation again:
[tex]\[ g = k \cdot h \cdot j \][/tex]
Substitute [tex]\( k = 24 \)[/tex], [tex]\( h = 2 \)[/tex], and [tex]\( j = 3 \)[/tex]:
[tex]\[ g = 24 \cdot 2 \cdot 3 \][/tex]
Simplify the expression:
[tex]\[ g = 24 \cdot 6 \][/tex]
[tex]\[ g = 144 \][/tex]
So, the value of [tex]\( g \)[/tex] when [tex]\( h = 2 \)[/tex] and [tex]\( j = 3 \)[/tex] is [tex]\( \boxed{144} \)[/tex].
