Optimization — Question 3

Question 3 - Solution

Let: - $r$: radius of the cylinder - $h$: height of the cylinder

Volume constraint: $$V = \pi r^2 h = 1000 \quad \Rightarrow \quad h = \frac{1000}{\pi r^2}$$

Cost: - Side area = $2\pi r h$, unit cost = 1 - Top and bottom area = $2\pi r^2$, unit cost = 2

Total cost $C$ (proportional to material area): $$C = 2\pi r h + 2 \cdot 2\pi r^2 = 2\pi r h + 4\pi r^2$$

Substitute $h$: $$C(r) = 2\pi r \cdot \frac{1000}{\pi r^2} + 4\pi r^2 = \frac{2000}{r} + 4\pi r^2$$

Differentiate: $$C'(r) = -\frac{2000}{r^2} + 8\pi r$$

Set derivative to 0: $$-\frac{2000}{r^2} + 8\pi r = 0 \Rightarrow 8\pi r^3 = 2000 \Rightarrow r^3 = \frac{250}{\pi}$$

$$r = \sqrt[3]{\frac{250}{\pi}}, \quad h = \frac{1000}{\pi r^2}$$

(a) Optimal Dimensions: $$\boxed{r = \sqrt[3]{\frac{250}{\pi}}, \quad h = \frac{1000}{\pi r^2}}$$

(b) Minimum Cost (in material area units): $$\boxed{C = \frac{2000}{r} + 4\pi r^2} \quad \text{where } r = \sqrt[3]{\frac{250}{\pi}}$$