Alright, let's tackle this problem step-by-step.
When a variable [tex]\( x \)[/tex] varies inversely as a variable [tex]\( t \)[/tex], it means that their product is a constant. This relationship can be written as:
[tex]\[ x \times t = k \][/tex]
where [tex]\( k \)[/tex] is a constant.
We are given:
- [tex]\( x = 8 \)[/tex] when [tex]\( t = 6 \)[/tex].
Using the inverse variation formula, we can determine the constant [tex]\( k \)[/tex]:
[tex]\[ 8 \times 6 = k \][/tex]
Calculating the value of [tex]\( k \)[/tex]:
[tex]\[ k = 48 \][/tex]
Now we need to find the value of [tex]\( t \)[/tex] when [tex]\( x = 12 \)[/tex].
Again, using the inverse variation formula:
[tex]\[ x \times t = k \][/tex]
Substitute [tex]\( x = 12 \)[/tex] and [tex]\( k = 48 \)[/tex] into the formula:
[tex]\[ 12 \times t = 48 \][/tex]
Solving for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{48}{12} \][/tex]
[tex]\[ t = 4 \][/tex]
So, when [tex]\( x = 12 \)[/tex], [tex]\( t = 4 \)[/tex].
Thus, the constant [tex]\( k \)[/tex] is [tex]\( 48 \)[/tex], and the value of [tex]\( t \)[/tex] when [tex]\( x = 12 \)[/tex] is [tex]\( 4.0 \)[/tex].
