optimization of sphere surface area formula with a fixed volume

To optimize the surface area of a sphere with a **fixed volume**, we need to analyze the relationship between the surface area and the volume, which involves using the following two formulas:

- **Surface Area of a sphere**: \( A = 4\pi r^2 \)
- **Volume of a sphere**: \( V = \frac{4}{3}\pi r^3 \)

### Step-by-step optimization process:

#### Step 1: Express \( r \) in terms of the volume
Given the volume is fixed, we want to express the radius \( r \) in terms of the volume \( V \). Starting from the volume formula:
\[
V = \frac{4}{3}\pi r^3
\]
Solving for \( r \):
\[
r = \left( \frac{3V}{4\pi} \right)^{\frac{1}{3}}
\]

#### Step 2: Substitute \( r \) into the surface area formula
Now that we have \( r \) in terms of \( V \), substitute this into the surface area formula \( A = 4\pi r^2 \):
\[
A = 4\pi \left( \left( \frac{3V}{4\pi} \right)^{\frac{1}{3}} \right)^2
\]
Simplify the expression:
\[
A = 4\pi \left( \frac{3V}{4\pi} \right)^{\frac{2}{3}}
\]
Thus, the surface area in terms of volume is:
\[
A = 4\pi \left( \frac{3^{2/3}V^{2/3}}{(4\pi)^{2/3}} \right)
\]
This can be further simplified:
\[
A = \left( 36\pi \right)^{1/3} V^{2/3}
\]

#### Step 3: Optimize the surface area
For a **fixed volume**, the formula shows that the surface area \( A \) depends on \( V^{2/3} \). Since the volume is constant, there's no further optimization required in terms of geometry, because a sphere minimizes surface area for a given volume compared to other shapes.

#### Conclusion:
The surface area is already optimized when the shape is a sphere, given a fixed volume. The formula \( A = \left( 36\pi \right)^{1/3} V^{2/3} \) represents the minimum surface area that a sphere can have for a fixed volume.

### Do you have any questions or want further details?

Here are five related questions:
1. What is the mathematical reasoning behind why a sphere minimizes surface area for a fixed volume?
2. How would the optimization change if we considered different shapes, like a cylinder or cube?
3. Can this formula be applied to non-Euclidean geometries (e.g., on a curved surface)?
4. How does this optimization relate to real-world problems, like minimizing material usage in packaging?
5. What role do the constants \( \pi \) and \( 36 \) play in the final optimized surface area formula?

**Tip**: The sphere has the most efficient surface area-to-volume ratio of any shape in three dimensions, which is why bubbles and planets tend to form spherical shapes!

Explore the process of optimizing the surface area of a sphere with a fixed volume. This detailed solution covers essential geometry and algebra concepts, including the surface area and volume formulas, and demonstrates how a sphere achieves the minimal surface area for a given volume. Suitable for high school students in Grades 10-12.

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