To find the scale factor of two similar solids when given the ratio of their volumes, we need to understand the relationship between the volumes of similar solids and their linear dimensions.
Here's the step-by-step solution:
1. Volume Ratio: The ratio of the volumes of the two solids is given as 27:1000. This can be written as a fraction:
[tex]\[
\text{Volume Ratio} = \frac{27}{1000}
\][/tex]
2. Understanding Scale Factor: For similar solids, the volume ratio is the cube of the scale factor. If we let the scale factor be [tex]\( k \)[/tex]:
[tex]\[
(\text{Scale Factor})^3 = \frac{27}{1000}
\][/tex]
3. Solving for the Scale Factor: To find the scale factor [tex]\( k \)[/tex], we need to take the cube root of the volume ratio. Mathematically, it can be expressed as:
[tex]\[
k = \sqrt[3]{\frac{27}{1000}}
\][/tex]
4. Compute the Cube Root:
- The cube root of 27 is 3, because [tex]\( 3^3 = 27 \)[/tex].
- The cube root of 1000 is 10, because [tex]\( 10^3 = 1000 \)[/tex].
Therefore,
[tex]\[
k = \frac{\sqrt[3]{27}}{\sqrt[3]{1000}} = \frac{3}{10}
\][/tex]
5. Simplify the Scale Factor:
[tex]\[
k = 0.3
\][/tex]
So, the scale factor is:
[tex]\[
\text{Scale Factor} = 0.3
\][/tex]
Alternatively, written in fraction form, the scale factor is [tex]\( \frac{3}{10} \)[/tex].
Hence, the scale factor between the two similar solids is [tex]\( 0.3 \)[/tex] or [tex]\( \frac{3}{10} \)[/tex].
