Answer :

To solve this problem, we need to understand the concept of inverse variation. When two variables, [tex]\(y\)[/tex] and [tex]\(x\)[/tex], vary inversely, it means that the product of [tex]\(y\)[/tex] and [tex]\(x\)[/tex] is always a constant value [tex]\(k\)[/tex]. Mathematically, this relationship is expressed as:

[tex]\[ y \cdot x = k \][/tex]

### Step-by-Step Solution:

1. Identify the constant [tex]\(k\)[/tex]:
We are given that [tex]\(y = 16\)[/tex] when [tex]\(x = 3\)[/tex]. Using the inverse variation formula, we can find the constant [tex]\(k\)[/tex].

[tex]\[ y \cdot x = k \implies 16 \cdot 3 = k \][/tex]

[tex]\[ k = 48 \][/tex]

So, the constant [tex]\(k\)[/tex] is 48.

2. Use the constant [tex]\(k\)[/tex] to find the new value of [tex]\(y\)[/tex] when [tex]\(x = 21\)[/tex]:
Now we need to find the value of [tex]\(y\)[/tex] when [tex]\(x = 21\)[/tex]. Using the inverse variation relationship [tex]\(y \cdot x = k\)[/tex], we substitute the known values:

[tex]\[ y \cdot 21 = 48 \][/tex]

To find [tex]\(y\)[/tex], solve the equation for [tex]\(y\)[/tex]:

[tex]\[ y = \frac{48}{21} \][/tex]

Simplify the fraction:

[tex]\[ y \approx 2.2857142857142856 \][/tex]

Therefore, when [tex]\(x = 21\)[/tex], the value of [tex]\(y\)[/tex] is approximately [tex]\(2.2857142857142856\)[/tex].