To solve the optimization problem for the cylindrical beverage can:

Problem Breakdown:

• Objective: Minimize the surface area of the cylinder (to minimize sheet metal used).
• Constraint: The cylinder must have a volume of 0.5 liters, i.e., $V = 0.5 \, \text{L} = 500 \, \text{cm}^3$ (converting to cm³).
• Unknowns: Height $h$ and radius $r$ of the cylinder.

Step 1: Expressions for Volume and Surface Area

1. Volume: $V = \pi r^2 h = 500$ Solve for $h$: $h = \frac{500}{\pi r^2}$

2. Surface Area: The surface area includes the side and two circular ends: $S = 2\pi r h + 2\pi r^2$ Substitute $h = \frac{500}{\pi r^2}$ into $S$: $S = 2\pi r \left(\frac{500}{\pi r^2}\right) + 2\pi r^2$ Simplify: $S = \frac{1000}{r} + 2\pi r^2$

Step 2: Minimize Surface Area

To minimize $S$, take the derivative $\frac{dS}{dr}$ and set it to zero.

1. Differentiate: $\frac{dS}{dr} = -\frac{1000}{r^2} + 4\pi r$

2. Set $\frac{dS}{dr} = 0$: $-\frac{1000}{r^2} + 4\pi r = 0$ Rearrange: $\frac{1000}{r^2} = 4\pi r$ Multiply through by $r^2$: $1000 = 4\pi r^3$ Solve for $r^3$: $r^3 = \frac{1000}{4\pi} = \frac{250}{\pi}$ Take the cube root: $r = \sqrt[3]{\frac{250}{\pi}}$

3. Approximate $r$ (using $\pi \approx 3.1416$): $r \approx \sqrt[3]{79.5775} \approx 4.31 \, \text{cm}$

Step 3: Calculate $h$

Using $h = \frac{500}{\pi r^2}$: $h = \frac{500}{\pi (4.31)^2}$ $h \approx \frac{500}{3.1416 \cdot 18.57} \approx \frac{500}{58.35} \approx 8.57 \, \text{cm}$

Final Answer:

• Radius: $r \approx 4.31 \, \text{cm}$
• Height: $h \approx 8.57 \, \text{cm}$

Would you like detailed steps for approximations or further clarifications?

Related Questions:

1. What happens to the dimensions if the volume is increased to 1 liter?
2. How does the shape of the can affect material costs in production?
3. Can this optimization process be extended to include thermal insulation requirements?
4. How does using a different material for the can influence the optimization constraints?
5. What are the trade-offs between minimizing surface area and maximizing stackability?

Tip:

Always convert all measurements to the same unit (e.g., cm³) before solving optimization problems.