A laundry soap company decides it wants to shrink the size of its box to save on cost. The surface area of the original box is 112 cm2 . If it creates a new box using a scale factor of 34 , determine the surface area of the new box. Round to the nearest tenth if necessary.(1 point)

Answer 1

To determine the surface area of the new box after applying a scale factor, we use the formula for the surface area scaling:

\[ \text{Surface Area of New Box} = k^2 \times \text{Surface Area of Original Box} \]

where \( k \) is the scale factor.

In this case, the scale factor \( k \) is \( \frac{1}{34} \) since the box is being shrunk (the scale factor is less than 1).

The original surface area is \( 112 , \text{cm}^2 \).

Now, calculating the surface area of the new box:

\[ \text{Surface Area of New Box} = \left(\frac{1}{34}\right)^2 \times 112 \]

Calculating \( \left(\frac{1}{34}\right)^2 \):

\[ \left(\frac{1}{34}\right)^2 = \frac{1}{1156} \]

Now substitute this back into the surface area formula:

\[ \text{Surface Area of New Box} = \frac{1}{1156} \times 112 \]

Calculating:

\[ \text{Surface Area of New Box} = \frac{112}{1156} \]

Calculating \( \frac{112}{1156} \):

\[ \text{Surface Area of New Box} \approx 0.0968 , \text{cm}^2 \]

Rounding to the nearest tenth:

\[ \text{Surface Area of New Box} \approx 0.1 , \text{cm}^2 \]

Thus, the surface area of the new box is approximately 0.1 cm².