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].
