To determine the surface area of the new box based on the scale factor, we use the formula for the surface area when the size of the object is scaled by a factor.
If the scale factor is \( k \), the surface area of the new box can be calculated using the following formula:
\[ \text{New Surface Area} = k^2 \times \text{Original Surface Area} \]
Given that the original surface area is 112 cm² and the scale factor is \( \frac{1}{34} \) (since it shrinks the size), we first calculate \( k^2 \):
\[ k = \frac{1}{34} \] \[ k^2 = \left( \frac{1}{34} \right)^2 = \frac{1}{1156} \]
Now we can find the new surface area:
\[ \text{New Surface Area} = \frac{1}{1156} \times 112 \]
Calculating this gives:
\[ \text{New Surface Area} = \frac{112}{1156} \approx 0.0967 \text{ cm}^2 \]
Rounding to the nearest tenth, the surface area of the new box is:
\[ \text{New Surface Area} \approx 0.1 \text{ cm}^2 \]
Thus, the surface area of the new box is approximately 0.1 cm².