When a three-dimensional shape such as a rectangular prism is dilated by a scale factor \(k\), the surface area of the new shape can be calculated using the formula:
\[ \text{New Surface Area} = k^2 \times \text{Original Surface Area} \]
In this case, the scale factor \(k\) is 3, and the original surface area is \(4 , \text{in}^2\).
First, calculate \(k^2\):
\[ k^2 = 3^2 = 9 \]
Now, multiply the original surface area by \(k^2\):
\[ \text{New Surface Area} = 9 \times 4 , \text{in}^2 = 36 , \text{in}^2 \]
Therefore, the surface area of the new dilated rectangular prism is \(36 , \text{in}^2\).