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: $$k^3=\frac{1728÷512}{512÷512}=\frac{3.375}{1}=3.375$$

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

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