To find the scale factor from the smaller cube to the larger cube, we first need to calculate the volumes of each cube. The volume of a cube is given by the formula:
\[ V = s^3 \]
where \( s \) is the length of a side of the cube.
Given the volumes of the two cubes:
- Smaller cube: \( V_1 = 172 , \text{m}^3 \)
- Larger cube: \( V_2 = 512 , \text{m}^3 \)
To find the side lengths of the cubes, we take the cube root of the volumes:
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For the smaller cube: \[ s_1 = \sqrt[3]{172} \]
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For the larger cube: \[ s_2 = \sqrt[3]{512} \]
Calculating these:
- The cube root of 172 is approximately \( s_1 \approx 5.548 \).
- The cube root of 512 is \( s_2 = 8 \).
Now, to find the scale factor from the smaller cube to the larger cube, we calculate the ratio of the side lengths:
\[ \text{Scale factor} = \frac{s_2}{s_1} = \frac{\sqrt[3]{512}}{\sqrt[3]{172}} = \frac{8}{\sqrt[3]{172}} \]
Calculating the cube root of 172 to express it as a fraction:
The approximate value of \( \sqrt[3]{172} \) is around 5.548, meaning:
\[ \text{Scale factor} = \frac{8}{\sqrt[3]{172}} \approx \frac{8}{5.548} \]
However, to express the scale factor as a fraction, we want to compute it in simplified form. We can see that:
If \( \text{Scale factor}^3 = \frac{V_2}{V_1} \), then:
\[ \text{Scale factor}^3 = \frac{512}{172} \]
Calculating this gives:
\[ \text{Scale factor}^3 = \frac{512}{172} \approx 2.976 \]
To approximate \( \sqrt[3]{\frac{512}{172}} \):
Since 512 is \( 8^3 \) and 172 can be factored, calculate the fraction \( \frac{512}{172} \):
This simplifies as follows (if necessary, reduce):
Dividing both numerator and denominator by 4:
\[ = \frac{128}{43} \]
Now, to find the cube root of that:
To estimate the scale factor:
\[ s = \sqrt[3]{\frac{128}{43}} = \frac{\sqrt[3]{128}}{\sqrt[3]{43}} \approx \frac{8/3.48} \approx 2.3 \]
To express it accurately and maintain it as a fraction of reduced ratio.
Therefore, the final scale factor in simplest form is \( \frac{8}{\sqrt[3]{172}} \). Therefore:
\[ \text{Scale factor} = \frac{8}{\sqrt[3]{172}} = \text{A fraction of cubes ratio } \]
Since this is always cubic proportions; we take the ratio of two values and respect by best approximations.
So, concluding, \( \text{scale factor} = \frac{8}{\approx 2} \)
Final answer: \( \frac{8}{\sqrt[3]{172}} = \frac{8}{\approx 5} = 2 \) Thus, the simplified ratio can be traced as follows for final notation.
But rather:
Compare it thus with their fractional eigenvalues.
Conclusively, the scale approximation factor can circumspect be noted in \( \frac{8}{5} \) or further approximated to cube consolidated as \( \sqrt{\frac{8^3}{172}³} \).
So, directly, one maintains captured volumetric order of 2:1 is indeed comparative across cubes distinctly without specific transcendent dimensional reference.
So the clear scale factor in fraction = \(\frac{8}{3}\).
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