Optimizing Cylinder Design for Minimized Surface Area Given a Fixed Volume

This article explores the mathematical optimization of a cylinder's design to minimize its surface area while maintaining a fixed volume. We will derive the formulas and go through the steps needed to find the optimal dimensions for such a cylinder.

Introduction

A cylinder is a common three-dimensional shape with numerous applications, ranging from engineering to packaging design. However, optimizing its shape for practical purposes involves balancing various factors, such as minimizing the surface area for cost-effective manufacturing. This article will guide you through the process of minimizing the surface area of a cylinder given a fixed volume of 5 cubic meters (m3).

The Volume Constraint

The volume of a cylinder can be calculated using the formula:

Volume (V) πr2h

Given that the volume is 5 m3, we can express the height (h) in terms of the radius (r):

h 5/πr2

The Surface Area of a Cylinder

The surface area (SA) of a cylinder can be expressed as:

SA 2πrh 2πr2

Since we already have h in terms of r, we substitute it into the surface area equation:

SA 2πr(5/πr2) 2πr2

Simplifying the function:

SA 10/r 2πr2

Optimizing the Surface Area

To find the minimum surface area, we need to take the derivative of SA with respect to r and set it to zero:

SA' d(10/r 2πr2) / dr

Using the derivative rules:

SA' -10/r2 4πr

Solving for the critical points:

-10/r2 4πr 0

4πr 10/r2

r3 10/4π

r (10/4π)1/3

Calculating the Optimal Radius and Surface Area

Substitute the optimal radius back into the surface area formula:

SA 10/r 2πr2

SA 10/(10/4π)1/3 2π(10/4π)2/3

Approximating the value:

SA ≈ 16.1868 m2

Conclusion

In summary, the minimum surface area of a cylinder with a fixed volume of 5 m3 is approximately 16.1868 m2. This occurs when the radius is given by the value we derived, (10/4π)1/3. The step-by-step process used to determine this minimum surface area is crucial for optimizing cylinder design in various applications.

Keywords

cylinder surface area, minimum surface area, volume constraint