To find the area of a rectangular prism, we need to first understand that the area in question likely refers to the surface area of the prism, but we can also determine the volume and dimensions from the given information.
The volume \( V \) of a rectangular prism is given by the formula:
\[ V = \text{length} \times \text{width} \times \text{height} \]
Given:
- Volume \( V = 235.98 , \text{cm}^3 \)
- Length \( l = 9 , \text{cm} \)
- Height \( h = 4.6 , \text{cm} \)
We need to find the width \( w \) first. Rearranging the volume formula to solve for width gives:
\[ w = \frac{V}{l \times h} \]
Substituting in the known values:
\[ w = \frac{235.98 , \text{cm}^3}{9 , \text{cm} \times 4.6 , \text{cm}} \]
Calculating the denominator:
\[ 9 , \text{cm} \times 4.6 , \text{cm} = 41.4 , \text{cm}^2 \]
Now substituting this into the equation for \( w \):
\[ w = \frac{235.98}{41.4} \approx 5.69 , \text{cm} \]
Now that we have the dimensions of the rectangular prism:
- Length \( l = 9 , \text{cm} \)
- Width \( w \approx 5.69 , \text{cm} \)
- Height \( h = 4.6 , \text{cm} \)
Now we can calculate the surface area \( SA \) using the formula:
\[ SA = 2(lw + lh + wh) \]
Calculating each term:
- \( lw = 9 , \text{cm} \times 5.69 , \text{cm} \approx 51.21 , \text{cm}^2 \)
- \( lh = 9 , \text{cm} \times 4.6 , \text{cm} = 41.4 , \text{cm}^2 \)
- \( wh \approx 5.69 , \text{cm} \times 4.6 , \text{cm} \approx 26.21 , \text{cm}^2 \)
Adding these together:
\[ lw + lh + wh \approx 51.21 + 41.4 + 26.21 \approx 118.82 , \text{cm}^2 \]
Now calculating the surface area \( SA \):
\[ SA = 2 \times 118.82 \approx 237.64 , \text{cm}^2 \]
Therefore, the surface area of the rectangular prism is approximately:
\[ \boxed{237.64 , \text{cm}^2} \]