Question

Two similar boxes are used for packaging laundry soap. One box has a volume of 648 inches ^(3) ; the other box has volume of 24 inches ^(3) . Compare the volumes and determine the scale factor between the two boxes.

Answer 1

To find the scale factor between the two boxes, we can compare their volumes. The scale factor for similar shapes can be determined by taking the cube root of the ratio of their volumes.

Given the volumes of the two boxes:

- Volume of the larger box = 648 inches³
- Volume of the smaller box = 24 inches³

First, we calculate the ratio of the volumes:

\[
\text{Volume ratio} = \frac{\text{Volume of larger box}}{\text{Volume of smaller box}} = \frac{648}{24}
\]

Calculating that:

\[
\frac{648}{24} = 27
\]

Next, we need to find the cube root of this ratio to determine the scale factor:

\[
\text{Scale factor} = \sqrt[3]{27} = 3
\]

Thus, the scale factor between the two boxes is \( \boxed{3} \).