To solve this problem, we start with the knowledge that [tex]\( x \)[/tex] is inversely proportional to the square root of [tex]\( y \)[/tex]. This relationship can be expressed as:
[tex]\[ x = \frac{k}{\sqrt{y}} \][/tex]
where [tex]\( k \)[/tex] is a constant that we need to determine.
1. Find the constant [tex]\( k \)[/tex]:
We are given that when [tex]\( x = 12 \)[/tex], [tex]\( y = 9 \)[/tex]. We can use these values to find [tex]\( k \)[/tex].
[tex]\[ 12 = \frac{k}{\sqrt{9}} \][/tex]
The square root of 9 is 3, so we substitute that into the equation:
[tex]\[ 12 = \frac{k}{3} \][/tex]
To solve for [tex]\( k \)[/tex], we multiply both sides by 3:
[tex]\[ k = 12 \times 3 = 36 \][/tex]
So, the constant [tex]\( k \)[/tex] is 36.
2. Find the new value of [tex]\( x \)[/tex] when [tex]\( y = 81 \)[/tex]:
Now that we have [tex]\( k \)[/tex], we use it to find the value of [tex]\( x \)[/tex] when [tex]\( y = 81 \)[/tex].
[tex]\[ x = \frac{36}{\sqrt{81}} \][/tex]
The square root of 81 is 9, so we substitute that into the equation:
[tex]\[ x = \frac{36}{9} \][/tex]
To solve for [tex]\( x \)[/tex], we simply divide 36 by 9:
[tex]\[ x = 4 \][/tex]
So, the value of [tex]\( x \)[/tex] when [tex]\( y = 81 \)[/tex] is [tex]\( \boxed{4} \)[/tex].
