Finding the Ratio of Radii of Similar Cylinders Based on Volume Ratio
Finding the Ratio of Radii of Similar Cylinders Based on Volume Ratio
When dealing with similar cylinders, the relationship between their volumes, radii, and heights can provide valuable insights. In this article, we will explore a step-by-step solution to determine the ratio of the radii of two given similar cylinders based on their volume ratios. The given volumes are 594 cubic centimeters and 4752 cubic centimeters, respectively.
Understanding Similar Cylinders and Volume Ratios
Two cylinders are said to be similar if their corresponding dimensions are proportional. In simpler terms, if the ratio of the radii of two similar cylinders is the same as the ratio of their heights, then the cylinders are similar. This property plays a crucial role in determining volume ratios between similar objects.
Given Data
Cylinder 1 (C1) Volume: 594 cm3 Cylinder 2 (C2) Volume: 4752 cm3The goal is to find the ratio of the radii of the smaller cylinder (C1) to the larger cylinder (C2).
Plan and Solution
The volumes of similar cylinders are proportional to the cube of their corresponding linear dimensions (radii, heights, etc.). Therefore, the ratio of the volumes can be expressed as the cube of the ratio of the radii (or heights).
Step 1: Calculate the ratio of the volumes.
V1 / V2 594 / 4752
Simplifying the ratio, we get:
594 / 4752 1 / 8
Step 2: Use the relationship between volume and radius for similar cylinders.
Since the cylinders are similar, the ratio of the radii will be the cube root of the ratio of the volumes.
3√(V1 / V2) 3√(1 / 8) 1 / 2
Conclusion: The ratio of the radii of the smaller cylinder to the larger cylinder is 1:2. This means the radius of the smaller cylinder is half the radius of the larger cylinder.
Step-by-Step Solution with Formulas
The volume of a cylinder is given by the formula:
V πr2 x h
When the heights are the same, the volume ratio simplifies to the ratio of the radii squared:
V1 / V2 r12 / r22
Given:
V1 594 cm3V2 4752 cm3
Therefore:
594 / 4752 r12 / r22
As before, simplifying the ratio, we find:
594 / 4752 1 / 8
Thus:
r1 / r2 1 / 2
Conclusion
The ratio of the radii of the smaller cylinder to the larger cylinder is 1:2. This result is a direct application of the proportional relationships between the volumes and the radii of similar cylinders.
To solve similar problems, always start by establishing the given data, using the appropriate formulas, and simplifying the ratios accordingly.
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