Question

The two cubes are similar in shape. Compare the volume of the two and determine the scale factor from the smaller cube to the larger cube. Enter your response as a fraction.
v =512m^2
v=1728m^3

Answer 1

To determine the scale factor between the two similar cubes based on their volumes, we first need to find the volume of each cube in a consistent unit.

The volumes given are:
- Volume of the smaller cube: \( V_1 = 512 \, \text{m}^3 \)
- Volume of the larger cube: \( V_2 = 1728 \, \text{m}^3 \)

The scale factor from the smaller cube to the larger cube can be calculated using the formula related to the volumes of similar shapes. If the scale factor is \( k \), then the relationship between the volumes of similar shapes is given by:

\[
\frac{V_2}{V_1} = k^3
\]

Now we can find the scale factor:

\[
k^3 = \frac{V_2}{V_1} = \frac{1728}{512}
\]

Simplifying the fraction:

\[
\frac{1728 \div 512}{512 \div 512} = \frac{3.375}{1} = \frac{27}{8}
\]

Now, we take the cube root of both sides to find \( k \):

\[
k = \sqrt[3]{\frac{27}{8}} = \frac{\sqrt[3]{27}}{\sqrt[3]{8}} = \frac{3}{2}
\]

Thus, the scale factor from the smaller cube to the larger cube is:

\[
\frac{3}{2}
\]

So the final answer is:

\[
\frac{3}{2}
\]