Packaging design is a real-world optimization problem where math helps engineers reduce material, cost, and waste. For a cylindrical can, the goal is often to hold a fixed volume while using the least possible surface area of aluminum. Students can model the can with radius r and height h, then use geometry and calculus to find the best dimensions.
This project connects derivatives, measurement, graphing, and sustainability in a way that looks like an authentic STEM design challenge.
The key idea is to write surface area as a function of one variable by using the fixed volume equation. For a closed cylinder, volume is V = pi r^2 h and surface area is S = 2 pi r^2 + 2 pi r h. Substituting h = V/(pi r^2) into the surface area formula gives S(r) = 2 pi r^2 + 2V/r, which can be minimized using a derivative.
The calculus result shows that the most material-efficient closed cylinder has h = 2r, meaning the height equals the diameter.
Key Facts
- Cylinder volume: V = pi r^2 h.
- Closed cylinder surface area: S = 2 pi r^2 + 2 pi r h.
- For fixed volume, h = V/(pi r^2).
- Surface area as one variable: S(r) = 2 pi r^2 + 2V/r.
- Derivative for optimization: S'(r) = 4 pi r - 2V/r^2.
- Minimum surface area occurs when h = 2r, so the height equals the diameter.
Vocabulary
- Optimization
- Optimization is the process of finding the best value of a quantity, such as the minimum surface area for a fixed volume.
- Constraint
- A constraint is a condition that must stay true in a problem, such as a can needing to hold a fixed volume.
- Surface Area
- Surface area is the total outside area of a three-dimensional object, including the top, bottom, and curved side of a closed can.
- Derivative
- A derivative measures how fast a function changes and can identify where a maximum or minimum may occur.
- Critical Point
- A critical point is an input value where the derivative is zero or undefined and where an optimum may occur.
Common Mistakes to Avoid
- Forgetting the top and bottom circles makes the surface area too small. A closed can uses S = 2 pi r^2 + 2 pi r h, not just the side area.
- Treating radius and height as independent after fixing volume is wrong. The volume constraint means changing r forces h to change.
- Setting the surface area formula equal to zero does not find the minimum. You must minimize S by setting the derivative S'(r) equal to zero.
- Confusing height with diameter leads to the wrong ratio. The optimal result is h = 2r, which means height equals diameter, not height equals radius.
Practice Questions
- 1 A closed cylindrical can must hold 500 cm^3. Use r = (V/(2 pi))^(1/3) and h = 2r to find the optimal radius and height to the nearest tenth of a centimeter.
- 2 A can has radius 4 cm and height 10 cm. Calculate its volume and total surface area using V = pi r^2 h and S = 2 pi r^2 + 2 pi r h.
- 3 Many real cans do not have h = 2r exactly. Explain two practical reasons a company might choose a non-optimal surface area ratio, such as stacking strength, branding, shipping, or manufacturing limits.