Exploring the Equality of Curved Surface Area and Volume in Right Circular Cylinders

One of the fascinating mathematical relationships in geometry involves the equality of the curved surface area of a right circular cylinder and its volume. This article will explore the conditions under which this equality holds true and determine the numerical value of the radius of the base of the cylinder.

Formulas and Definitions

Let's start by recalling the formulas for the curved surface area (S) and volume (V) of a right circular cylinder:

Curved surface area (S) 2πRH Volume (V) πR2H

Equality of Curved Surface Area and Volume

The problem at hand is to determine the radius (R) of the base of a right circular cylinder for which the numerical value of the curved surface area is equal to the numerical value of its volume.

Mathematical Approach

If the numerical values of the curved surface area and the volume are equal, we can set the two formulas equal to each other:

[2πRH πR^2H]

Let's solve this equation step by step:

First, we can cancel out the common terms π and H from both sides of the equation (assuming H is not zero). This simplifies the equation to: [2R R^2] We can further simplify by rearranging the equation: [R^2 - 2R 0] This can be factored as: [R(R - 2) 0] From this factored form, we get two possible solutions: [R 0 quad text{or} quad R 2]

However, a radius of 0 would imply a degenerate cylinder (a line segment), and in this case, both the volume and the curved surface area would be zero. Therefore, the radius R cannot be 0 for a non-degenerate cylinder.

Conclusion

The only valid solution for the radius R of the base of the cylinder, under the condition that the numerical value of its curved surface area is equal to the numerical value of its volume, is:

[R 2]

This means that for a right circular cylinder, the numerical value of the radius of the base that satisfies the condition where the curved surface area is equal to the volume is 2 units.