Volume Calculation of Similar Cylinders: A Comprehensive Guide

Introduction

This article delves into the calculation of the volume of cylinder B, given the dimensions and properties of cylinder A. The focus is on understanding and applying the concept of mathematical similarity to solve geometric problems involving cylindrical shapes.

Problem Statement

The problem involves two cylinders, A and B. Cylinder A has a height of 5 cm and a cross-sectional area of 18 cm2. Cylinder B is mathematically similar to cylinder A and has a height of 10 cm. We need to determine the volume of cylinder B.

Method 1: Direct Calculation Using Similarity Ratios

1. **Identify the similar properties of the cylinders**:

Cylinder A:

Cylinder B:

Since the cylinders are similar, the ratio of their corresponding dimensions is constant.

Step 1:

Calculate the cross-sectional area of cylinder B.

Given that cylinder B is similar to cylinder A, the ratio of the heights is equal to the ratio of the radii:

[ frac{h_A}{h_B} frac{r_A}{r_B} frac{5}{10} frac{1}{2} ]

Therefore:

[ r_B 2 r_A ]

Calculate the area of the cross-section of cylinder B:

[ text{Area of cross-section of cylinder B} pi r_B^2 4 times pi r_A^2 4 times 18 72 text{ cm}^2 ]

Step 2: Calculate the volume of cylinder B:

[ V_B text{Area of cross-section} times text{Height of cylinder B} 72 times 10 720 text{ cm}^3 ]

Method 2: Using the Volume Ratio of Similar Solids

Another approach to solving this problem involves using the volume ratio of similar solids.

The volume of cylinder A (VA) is given by:

[ V_A 18 text{ cm}^2 times 5 text{ cm} 90 text{ cm}^3 ]

Since the cylinders are mathematically similar, the ratio of the volumes is equal to the cube of the ratio of their corresponding dimensions:

[ frac{V_A}{V_B} left( frac{h_A}{h_B} right)^3 left( frac{5}{10} right)^3 frac{1}{8} ]

Step 3: Calculate the volume of cylinder B:

[ V_B V_A times 8 90 times 8 720 text{ cm}^3 ]

Second Method: Direct Cross-Section Comparison

This method involves analyzing the cross-sectional area and height, rather than the radius.

The cross-sectional area of cylinder A is given by:

[ text{Area of cross-section of cylinder A} 18 text{ cm}^2 ]

Since cylinder B is similar to cylinder A, the ratio of their cross-sectional areas is equal to the square of the ratio of their heights:

[ frac{text{Area of cross-section of cylinder A}}{text{Area of cross-section of cylinder B}} left( frac{5}{10} right)^2 frac{1}{4} ]

Step 4: Calculate the cross-sectional area of cylinder B:

[ text{Area of cross-section of cylinder B} 18 times 4 72 text{ cm}^2 ]

Step 5: Calculate the volume of cylinder B:

[ V_B text{Area of cross-section} times text{Height of cylinder B} 72 times 10 720 text{ cm}^3 ]

Conclusion

Both methods yield the same result, confirming the calculation of the volume of cylinder B as 720 cm3. This example demonstrates the importance of understanding mathematical similarity and the application of geometric principles in solving problems related to similar solids.

By understanding these concepts, one can approach similar geometric problems effectively, ensuring accurate and efficient calculations.