Without using calculus, we can still reason about the relationship between the radius (\(r\)) and height (\(h\)) of a cylinder that minimizes its surface area by considering the properties of the cylinder and using basic geometric reasoning.

### Surface Area and Volume of a Cylinder
- The **surface area** \(S\) of a cylinder is composed of two parts:
  \[
  S = 2\pi r^2 + 2\pi rh
  \]
  - The first term, \(2\pi r^2\), is the area of the two circular bases.
  - The second term, \(2\pi rh\), is the area of the curved side (lateral surface).

- The **volume** \(V\) of the cylinder is:
  \[
  V = \pi r^2 h
  \]
  Here, the volume is fixed, meaning that as the radius increases, the height must decrease to keep the volume constant, and vice versa.

### Reasoning to Minimize Surface Area

1. **Balance Between Height and Radius**: 
   - If \(r\) is very small, then to maintain the fixed volume, \(h\) must be large. In this case, the lateral surface area (\(2\pi rh\)) becomes large, and the base area (\(2\pi r^2\)) is small.
   - If \(r\) is very large, \(h\) must be small. Now, the base area is large, but the lateral surface area becomes smaller.

2. **Optimal Shape (Visual Reasoning)**:
   - Think about the extreme cases: 
     - If the cylinder is very tall and skinny (large \(h\), small \(r\)), the lateral surface dominates, and the surface area is large.
     - If the cylinder is very short and wide (small \(h\), large \(r\)), the surface area of the bases dominates, and the surface area is again large.
   - The optimal situation is a balance where neither the base area nor the lateral surface area overly dominates.

3. **Symmetry and Geometric Intuition**:
   - A symmetric balance suggests that for a given volume, the most efficient (minimal surface area) shape would be one where the height \(h\) and the diameter \(2r\) are equal.
   - Since \(h\) equals the diameter \(2r\) means:
     \[
     h = 2r
     \]
   This relationship intuitively minimizes the surface area because it balances the contribution of the base area and the lateral surface area.

### Conclusion:
Without using calculus, we can conclude that the cylinder's surface area is minimized when the height \(h\) is equal to twice the radius \(r\), or \(h = 2r\). This is derived from reasoning about how the surface area changes with different proportions of height and radius, aiming to balance the areas contributed by the cylinder's sides and bases.

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Would you like more information or have any questions about this explanation? Here are some related questions you might find interesting:

1. How does the surface area compare for cylinders with different ratios of \(h\) and \(r\)?
2. Why is symmetry important in minimizing surface area?
3. What happens if the cylinder’s radius is fixed instead of its volume?
4. How does this reasoning apply to other shapes, like a sphere?
5. Can we find the minimum surface area of a cone without calculus?
6. How does this optimal relationship influence packaging design?
7. What would happen to the surface area if \(h\) were much larger than \(2r\)?
8. How does the formula for surface area change if the cylinder is hollow?

**Tip:** When optimizing shapes, looking for symmetry or balance often leads to the minimum or maximum values without needing advanced mathematics.

Understand how to minimize the surface area of a cylinder without using calculus. Learn about the relationship between radius and height, and the geometric reasoning behind minimizing surface area. Suitable for advanced high school and college-level students.

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