Optimize surface area of a plastic water bottle to find the best shape for achieving minimum casing material used.



To optimize the surface area of a plastic water bottle while maintaining a fixed volume, we aim to find the most efficient shape that minimizes the amount of plastic used for the casing. Here, we'll assume the bottle is cylindrical, as most commercial water bottles are.

### Step-by-step optimization process:

1. **Variables and Parameters**:
   - Let the radius of the cylindrical bottle be \( r \).
   - Let the height of the bottle be \( h \).
   - Assume the volume of the bottle is fixed at \( V \).

   The volume of the cylinder is given by:
   \[
   V = \pi r^2 h
   \]
   Rearranging to express \( h \) in terms of \( r \) and \( V \):
   \[
   h = \frac{V}{\pi r^2}
   \]

2. **Surface Area of the Bottle**:
   The total surface area of a cylindrical bottle (including the sides and two circular ends) is given by:
   \[
   A = 2\pi r^2 + 2\pi r h
   \]
   The first term \( 2\pi r^2 \) is the surface area of the two circular ends, and the second term \( 2\pi r h \) is the surface area of the side (the cylindrical part).

3. **Substitute \( h \) in Terms of \( r \)**:
   Using the expression for \( h \), substitute into the surface area equation:
   \[
   A = 2\pi r^2 + 2\pi r \left( \frac{V

Optimize surface area of a plastic water bottle to find the best shape for achieving minimum casing material used.

Explore how to optimize the surface area of a plastic water bottle to minimize casing material using cylindrical volume and surface area formulas. This problem includes geometry and optimization concepts, and is suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation