To determine the surface area of the new box after applying the scale factor of \( \frac{3}{4} \), we can use the formula for the surface area of similar objects, which states that the surface area scales with the square of the linear dimensions.
- Given the original surface area \( A_{original} = 112 , \text{cm}^2 \).
- The scale factor for the linear dimensions is \( \frac{3}{4} \).
The new surface area \( A_{new} \) can be calculated using the formula:
\[ A_{new} = A_{original} \times \left(\text{scale factor}\right)^2 \]
First, we calculate \( \left(\frac{3}{4}\right)^2 \):
\[ \left(\frac{3}{4}\right)^2 = \frac{9}{16} \]
Now, plug this value into the surface area formula:
\[ A_{new} = 112 , \text{cm}^2 \times \frac{9}{16} \]
Next, we multiply:
\[ A_{new} = 112 \times \frac{9}{16} = \frac{112 \times 9}{16} \]
Calculating \( 112 \times 9 \):
\[ 112 \times 9 = 1008 \]
Now, divide by 16:
\[ A_{new} = \frac{1008}{16} = 63 , \text{cm}^2 \]
Thus, the surface area of the new box is
\[ \boxed{63 , \text{cm}^2} \]