To find the missing width of the rectangular prism, we can use the formula for the surface area of a rectangular prism:
\[ \text{Surface Area} = 2(lw + lh + wh) \] where:
- \( l \) is the length,
- \( w \) is the width,
- \( h \) is the height.
Given:
- Length \( l = 6 , \text{cm} \)
- Height \( h = 4 , \text{cm} \)
- Surface Area \( = 188 , \text{cm}^2 \)
Now, we can plug in the known values:
\[ 188 = 2(6w + 6 \cdot 4 + 4w) \]
Simplifying inside the parentheses:
\[ 188 = 2(6w + 24 + 4w) \]
This simplifies to:
\[ 188 = 2(10w + 24) \]
Now, divide both sides by 2:
\[ 94 = 10w + 24 \]
Next, subtract 24 from both sides:
\[ 70 = 10w \]
Now, divide by 10:
\[ w = 7 , \text{cm} \]
So the missing width of the rectangular prism is \( 7 , \text{cm} \).