Question

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 120 cm?
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.
(1 point)
cm?

Answer 1

To find the surface area of the new box after applying a scale factor, we can use the relationship between the surface area and the scale factor.

The surface area of an object scales with the square of the scale factor. If we denote the original surface area as \( S_1 \) and the new surface area as \( S_2 \), we have:

\[
S_2 = S_1 \times \left( \text{scale factor} \right)^2
\]

Given:
- The original surface area \( S_1 = 120 \, \text{cm}^2 \)
- Scale factor = \( \frac{3}{4} \)

Now, we calculate the new surface area:

\[
S_2 = 120 \times \left( \frac{3}{4} \right)^2
\]

Calculating \( \left( \frac{3}{4} \right)^2 \):

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

Now substituting back into the equation for \( S_2 \):

\[
S_2 = 120 \times \frac{9}{16}
\]

Calculating \( S_2 \):

\[
S_2 = 120 \times 0.5625 = 67.5
\]

Thus, the surface area of the new box is:

\[
\text{Surface Area} = 67.5 \, \text{cm}^2
\]

Rounded to the nearest tenth:

\[
\text{Surface Area} = 67.5 \, \text{cm}^2
\]

So the answer is \( \text{67.5 cm}^2 \).