To find the missing width of a rectangular prism, we can use the formula for the surface area of a rectangular prism:
\[ \text{Surface Area} = 2lw + 2lh + 2wh \]
where:
- \( l \) is the length,
- \( w \) is the width,
- \( h \) is the height.
Given:
- \( \text{Surface Area} = 188 , \text{cm}^2 \)
- \( l = 6 , \text{cm} \)
- \( h = 4 , \text{cm} \)
We need to find \( w \).
Substituting the known values into the surface area formula:
\[ 188 = 2(6)(w) + 2(6)(4) + 2(4)(w) \]
Calculating \( 2(6)(4) \):
\[ 2(6)(4) = 48 \]
So, we can substitute that back into the equation:
\[ 188 = 12w + 48 + 8w \]
Now, combine like terms:
\[ 188 = 20w + 48 \]
Next, subtract 48 from both sides to isolate the term with \( w \):
\[ 188 - 48 = 20w \]
\[ 140 = 20w \]
Now, divide both sides by 20:
\[ w = \frac{140}{20} = 7 \]
Therefore, the missing width \( w \) is:
\[ \boxed{7 , \text{cm}} \]