Optimize surface area of a cylinder plastic bottle to achieve minimal casting materials used for 400 ml

To optimize the surface area of a cylinder (such as a plastic bottle) for a fixed volume, minimizing casting materials, we need to express the surface area as a function of the dimensions of the cylinder (radius and height) and then minimize it using calculus.

### 1. Variables and Formulas:
- Let the **radius** of the cylinder be \( r \).
- Let the **height** of the cylinder be \( h \).
- The **volume** of the cylinder is fixed at \( V = 400 \, \text{ml} = 400 \, \text{cm}^3 \).
  \[
  V = \pi r^2 h
  \]
- The **surface area** \( S \) (which includes both the lateral surface and the two circular ends) is given by:
  \[
  S = 2\pi r h + 2\pi r^2
  \]
  Here, the first term \( 2\pi r h \) is the lateral surface area, and the second term \( 2\pi r^2 \) represents the area of the two circular ends.

### 2. Express \( h \) in Terms of \( r \):
From the volume equation \( V = \pi r^2 h \), we can solve for \( h \):
\[
h = \frac{V}{\pi r^2} = \frac{400}{\pi r^2}
\]

### 3. Substitute \( h \) into the Surface Area Formula:
Substituting \( h = \frac{400}{\pi r^2} \) into the surface area formula \( S = 2\pi r h + 2\pi r^2 \), we get:
\[
S(r) = 2\pi r \left(\frac{400}{\pi r^2}\right) + 2\pi r^2
\]
Simplifying:
\[
S(r) = \frac{800}{r} + 2\pi r^2
\]

### 4. Minimize the Surface Area:
To minimize the surface area, we take the derivative of \( S(r) \) with respect to \( r \), set it equal to zero, and solve for \( r \).

\[
\frac{dS}{dr} = -\frac{800}{r^2} + 4\pi r
\]
Set \( \frac{dS}{dr} = 0 \):
\[
-\frac{800}{r^2} + 4\pi r = 0
\]
Solving for \( r \):
\[
\frac{800}{r^2} = 4\pi r
\]
\[
800 = 4\pi r^3
\]
\[
r^3 = \frac{800}{4\pi} = \frac{200}{\pi}
\]
\[
r = \left(\frac{200}{\pi}\right)^{1/3}
\]

Now, we calculate \( r \) and then find \( h \).

### 5. Calculate \( r \) and \( h \):

Let’s calculate \( r \) and substitute back to find \( h \).The optimized dimensions of the cylinder are:
- **Radius** \( r \approx 3.99 \, \text{cm} \)
- **Height** \( h \approx 7.99 \, \text{cm} \)

These dimensions minimize the surface area of the cylinder for a fixed volume of 400 ml, thus reducing the amount of material required for casting.

Would you like to see the surface area calculation or any additional details?

Here are 5 related questions:
1. How do these dimensions compare to standard bottle sizes?
2. What if the volume constraint were different (e.g., 500 ml)?
3. How does the material thickness affect the optimization?
4. Could the shape of the bottle impact strength or functionality despite surface area optimization?
5. How does this approach change if you optimize for weight instead of surface area?

**Tip:** When optimizing geometric shapes, expressing one variable in terms of the other often simplifies the process.

Optimize surface area of a cylinder plastic bottle to achieve minimal casting materials used for 400 ml.

Learn how to minimize the surface area of a cylindrical plastic bottle to reduce casting materials for a 400 ml volume. This step-by-step solution involves calculus and geometry, focusing on optimizing material use through surface area calculations.

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