To determine the surface area of the new box when the original box's surface area is given and a scale factor is applied, we can use the formula for scaling surface area.
When the dimensions of a three-dimensional object are scaled by a factor of \( k \), the surface area is scaled by a factor of \( k^2 \).
In this case, the original surface area is \( 112 , cm^2 \) and the scale factor provided is \( \frac{3}{4} \).
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First, calculate the square of the scale factor: \[ \left(\frac{3}{4}\right)^2 = \frac{9}{16} \]
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Next, multiply the original surface area by this scaling factor: \[ \text{New Surface Area} = 112 , cm^2 \times \frac{9}{16} \]
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Performing the multiplication: \[ 112 \times \frac{9}{16} = \frac{112 \times 9}{16} = \frac{1008}{16} = 63 \]
Therefore, the surface area of the new box is \( 63 , cm^2 \).
Thus, the answer is: \[ \text{Surface area of the new box: } 63 , cm^2 \]