Calculating the Radius Ratio of Similar Cylinders Based on Their Volumes

When dealing with similar three-dimensional shapes, the relationship between dimensions and volume can be mathematically deduced. In this article, we will explore how to find the ratio of the radii of two similar cylinders given their volumes. This type of problem is a common challenge in geometry and can be useful in various real-world applications, such as designing structures and models.

Problem Statement

We are given the volumes of two similar cylinders: the volume of the first cylinder is 594 cubic centimeters (cm3) and the volume of the second cylinder is 4752 cubic cm3.

Solution

Step 1: Calculate the Volume Ratio

The first step is to find the ratio of the volumes of the two cylinders:

[ frac{V_{1}}{V_{2}} frac{594}{4752} ]

Step 2: Simplify the Ratio

We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). In this case, the simplified ratio is:

[frac{594}{4752} frac{1}{8}]

Step 3: Apply the Volume Ratio to the Radii

For similar three-dimensional objects, the ratio of their volumes is equal to the cube of the ratio of their linear dimensions (such as radii) and heights. Therefore, if the volume ratio is 1:8, then the ratio of the radii must be the cube root of 1:8:

[sqrt[3]{frac{1}{8}} frac{1}{2}]

Hence, the ratio of the radius of the smaller cylinder to the radius of the larger cylinder is 1:2.

Step 4: Conclusion

The ratio of the radii of the smaller cylinder to the larger cylinder is 1:2. This means that the radius of the smaller cylinder is half the radius of the larger cylinder.

In mathematical notation:

[frac{r_{small}}{r_{big}} frac{1}{2}]

Conclusion

By understanding the relationship between the volumes of similar cylinders and their linear dimensions, we can easily find the ratio of their radii. This problem-solving approach is not only useful for academic purposes but can also be applied in practical situations where scaling and design are essential.

Understanding and applying these concepts can help in various fields such as engineering, architecture, and manufacturing. If you have similar problems or need further assistance, feel free to explore more resources and seek guidance.