To solve this problem, we need to use the relationship where [tex]\( m \)[/tex] varies inversely as the square of [tex]\( n \)[/tex]. This means that
[tex]\[
m = \frac{k}{n^2}
\][/tex]
for some constant [tex]\( k \)[/tex].
### Step 1: Find the constant [tex]\( k \)[/tex]
Given that [tex]\( m = 6 \)[/tex] when [tex]\( n = 3 \)[/tex], we can plug these values into the equation:
[tex]\[
6 = \frac{k}{3^2}
\][/tex]
Simplify [tex]\( 3^2 \)[/tex]:
[tex]\[
6 = \frac{k}{9}
\][/tex]
To find [tex]\( k \)[/tex], multiply both sides of the equation by 9:
[tex]\[
k = 6 \times 9 = 54
\][/tex]
So, the constant [tex]\( k \)[/tex] is 54.
### Step 2: Use [tex]\( k \)[/tex] to find the new value of [tex]\( n \)[/tex]
We are now given that [tex]\( m = 13.5 \)[/tex] and we need to find [tex]\( n \)[/tex]. Using the relationship [tex]\( m = \frac{k}{n^2} \)[/tex] again, we plug in the known values [tex]\( m = 13.5 \)[/tex] and [tex]\( k = 54 \)[/tex]:
[tex]\[
13.5 = \frac{54}{n^2}
\][/tex]
To solve for [tex]\( n^2 \)[/tex], multiply both sides by [tex]\( n^2 \)[/tex]:
[tex]\[
13.5 n^2 = 54
\][/tex]
Next, isolate [tex]\( n^2 \)[/tex] by dividing both sides by 13.5:
[tex]\[
n^2 = \frac{54}{13.5}
\][/tex]
Simplify the right side:
[tex]\[
n^2 = 4
\][/tex]
Now, take the positive square root of both sides to find the positive value of [tex]\( n \)[/tex]:
[tex]\[
n = \sqrt{4} = 2
\][/tex]
Therefore, the positive value of [tex]\( n \)[/tex] when [tex]\( m = 13.5 \)[/tex] is
[tex]\[
\boxed{2}
\][/tex]
