To compare the volumes of the two cubes and determine the scale factor, we start by noting their volumes:
- Volume of the smaller cube, \( V_1 = 512 , m^3 \)
- Volume of the larger cube, \( V_2 = 1728 , m^3 \)
To find the scale factor, we use the relationship between the volumes of similar shapes. The volume scale factor is the cube of the linear scale factor. If the linear scale factor is \( k \), then:
\[ \frac{V_2}{V_1} = k^3 \]
Calculating the ratio of the volumes:
\[ \frac{V_2}{V_1} = \frac{1728}{512} \]
Now we simplify this fraction:
- Divide both the numerator and denominator by 512:
\[ \frac{1728 \div 512}{512 \div 512} = \frac{3.375}{1} \]
Alternatively, we can calculate it step-by-step in integer terms. To simplify \( \frac{1728}{512} \):
Dividing both numerator and denominator by 64 (the greatest common divisor):
\[ \frac{1728 \div 64}{512 \div 64} = \frac{27}{8} \]
Thus, we have:
\[ \frac{V_2}{V_1} = \frac{27}{8} \]
Now, taking the cube root of both sides gives us the linear scale factor \( k \):
\[ k^3 = \frac{27}{8} \]
Taking the cube root:
\[ k = \sqrt[3]{\frac{27}{8}} = \frac{\sqrt[3]{27}}{\sqrt[3]{8}} = \frac{3}{2} \]
Therefore, the scale factor from the smaller cube to the larger cube, expressed as a fraction, is:
\[ \frac{3}{2} \]
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