Select the correct answer.

The value of [tex]$m$[/tex] varies inversely as the square of [tex]$n$[/tex]. When [tex]$n=3, m=6$[/tex]. What is the positive value of [tex][tex]$n$[/tex][/tex] when [tex]$m=13.5$[/tex]?

A. 2
B. 4
C. 18
D. 9



Answer :

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]