When [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex], we can express the relationship as:
[tex]\[ y = kx \][/tex]
where [tex]\( k \)[/tex] is the constant of variation.
Step 1: Find the constant of variation ([tex]\( k \)[/tex])
We are given that [tex]\( y = 40 \)[/tex] when [tex]\( x = 10 \)[/tex].
We substitute these values into the direct variation equation to find [tex]\( k \)[/tex]:
[tex]\[ 40 = k \cdot 10 \][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{40}{10} \][/tex]
[tex]\[ k = 4 \][/tex]
Step 2: Use the constant of variation to find [tex]\( x \)[/tex] when [tex]\( y \)[/tex] is 48
Now we know the value of [tex]\( k \)[/tex] is 4. We need to find [tex]\( x \)[/tex] when [tex]\( y = 48 \)[/tex].
We use the same direct variation equation:
[tex]\[ y = kx \][/tex]
Substitute [tex]\( y = 48 \)[/tex] and [tex]\( k = 4 \)[/tex]:
[tex]\[ 48 = 4x \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{48}{4} \][/tex]
[tex]\[ x = 12 \][/tex]
Therefore, when [tex]\( y \)[/tex] is 48, [tex]\( x \)[/tex] is 12.
So, [tex]\( x = \boxed{12} \)[/tex]
