To find the surface area of the new box after applying a scale factor of \( \frac{3}{4} \), we can use the property that when the linear dimensions of a shape are scaled by a factor \( k \), the surface area is scaled by \( k^2 \).
In this case, \( k = \frac{3}{4} \). Thus, the new surface area \( A' \) can be calculated as follows:
\[ A' = A \cdot k^2 \]
where \( A \) is the original surface area. Given that the original surface area is \( 120 \text{ cm}^2 \):
\[ A' = 120 \cdot \left( \frac{3}{4} \right)^2 \]
Calculating \( \left( \frac{3}{4} \right)^2 \):
\[ \left( \frac{3}{4} \right)^2 = \frac{9}{16} \]
Now substitute back into the equation for \( A' \):
\[ A' = 120 \cdot \frac{9}{16} \]
To simplify this calculation:
\[ A' = \frac{120 \cdot 9}{16} \]
Calculating \( 120 \cdot 9 = 1080 \):
\[ A' = \frac{1080}{16} \]
Now divide \( 1080 \) by \( 16 \):
\[ A' = 67.5 \]
Thus, the surface area of the new box is
\(\boxed{67.5} \text{ cm}^2\).
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