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- 512 m V - 172 m

Answer 1

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:

1. For the smaller cube:
\[
s_1 = \sqrt[3]{172}
\]

2. For the larger cube:
\[
s_2 = \sqrt[3]{512}
\]

Calculating these:

1. The cube root of 172 is approximately \( s_1 \approx 5.548 \).
2. 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 = **\(\frac{8}{3}\)**.