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. (1 point) V = 512m^3 for the small cube, and V = 1728 m^3 for the big cube

Answer 1

To find the scale factor between the two similar cubes based on their volumes, we can use the relationship between the volumes of similar shapes. The volume of a similar shape scales with the cube of the linear dimension scale factor \( k \).

1. **Finding the scale factor from the volumes**:

The formula relating the volume scale factor \( k^3 \) to the volumes \( V_1 \) and \( V_2 \) of two similar objects is:
\[
k^3 = \frac{V_2}{V_1}
\]

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

Now, substituting in the values:
\[
k^3 = \frac{1728}{512}
\]

2. **Calculating the fraction**:
\[
k^3 = \frac{1728 \div 512}{512 \div 512} = \frac{3.375}{1} = 3.375
\]

3. **Finding \( k \)**:
To find \( k \), take the cube root of \( k^3 \):
\[
k = \sqrt[3]{\frac{1728}{512}} = \sqrt[3]{\frac{1728 \div 512}{512 \div 512}} = \sqrt[3]{3.375}
\]
\[
k = \frac{\sqrt[3]{1728}}{\sqrt[3]{512}} = \frac{12}{8} = \frac{3}{2}
\]

Therefore, the scale factor from the smaller cube to the larger cube is:
\[
\boxed{\frac{3}{2}}
\]