If two cylinders are similar and the ratio between the lengths of their edges is 2:5, what is the ratio of their volumes?A.2:5B.4:25C.8:20D.8:125

1. Break Down the Problem

To solve the problem, we need to find the ratio of the volumes of two similar cylinders given the ratio of their corresponding lengths (edges). Since the cylinders are similar, the ratio of their volumes can be derived from the ratio of their linear dimensions (lengths).

2. Relevant Concepts

For similar geometric shapes, the ratio of their volumes is the cube of the ratio of their corresponding linear dimensions. If the ratio of the edges of the two cylinders is $a:b$, then the ratio of their volumes $V_1:V_2$ can be expressed as: $\frac{V_1}{V_2} = \left( \frac{a}{b} \right)^3$

3. Analysis and Detail

Given that the ratio of the lengths of the edges is $2:5$, we can assign:

• $a = 2$
• $b = 5$

Now, we will calculate the volume ratio: $\frac{V_1}{V_2} = \left( \frac{2}{5} \right)^3 = \frac{2^3}{5^3} = \frac{8}{125}$

4. Verify and Summarize

We have derived the ratio of the volumes based on the similarity of the cylinders and the given ratio of their lengths. The calculations confirm that the ratio of their volumes is $\frac{8}{125}$.

Final Answer

The ratio of the volumes of the two cylinders is $8:125$. Thus, the correct answer is D. 8:125.