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 cm^2 . If it creates a new box using a scale factor of 3/4 , determine the surface area of the new box. Round to the nearest tenth if necessary.

Question

(answer here) cm^2

Answer 1

The surface area of an object scales with the square of the scale factor. If the original surface area is \( S \) and the scale factor is \( k \), then the new surface area \( S' \) can be calculated using the formula:

\[ S' = S \times k^2 \]

In this case, the original surface area \( S = 112 , \text{cm}^2 \) and the scale factor \( k = \frac{3}{4} \). First, we calculate \( k^2 \):

\[ k^2 = \left( \frac{3}{4} \right)^2 = \frac{9}{16} \]

Now, we substitute the values into the formula for the new surface area:

\[ S' = 112 \times \frac{9}{16} \]

Calculating \( \frac{112 \times 9}{16} \):

First, calculate \( 112 \div 16 \):

\[ 112 \div 16 = 7 \]

Then, multiply this result by 9:

\[ S' = 7 \times 9 = 63 \]

Thus, the surface area of the new box is:

\[ \boxed{63} , \text{cm}^2 \]