To find the width \( w \) of the rectangular prism, we can rearrange the volume formula \( V = lwh \) to solve for \( w \):
\[ w = \frac{V}{lh} \]
Given:
- Volume \( V = 138.24 , \text{in}^3 \)
- Height \( h = 9.6 , \text{in} \)
- Length \( l = 3.2 , \text{in} \)
Now we can substitute the values into the equation:
\[ w = \frac{138.24}{3.2 \times 9.6} \]
First, calculate \( 3.2 \times 9.6 \):
\[ 3.2 \times 9.6 = 30.72 \]
Now substitute back into the width equation:
\[ w = \frac{138.24}{30.72} \]
Calculating that gives:
\[ w = 4.5 , \text{in} \]
Therefore, the width of the rectangular prism is \( \boxed{4.5} , \text{in} \).