An open water tank for a farm has a square base and vertical sides. It must hold 32 m³. Find the dimensions that need the least sheet metal, and that least area.

1. 1 · Write what is given and what is to be found. Square base of side x m, height h m, no lid, volume 32 m³. To find: x and h for the least surface area, and that area.
2. 2 · Draw the figure. Draw the open tank. Label the base x by x and the height h. Write V = 32 m³ beside it.
3. 3 · Write the formula, rule or theorem. Surface area S = x² + 4xh (base plus four sides). Condition: x²h = 32, so h = 32/x².
4. 4 · Write S in one variable. S = x² + 4x × 32/x² = x² + 128/x, x > 0
5. 5 · Differentiate and solve.
   • dS/dx = 2x − 128/x² = 0
   • x³ = 64, so x = 4
6. 6 · Use the second derivative test. d²S/dx² = 2 + 256/x³ = 2 + 4 = 6 > 0 at x = 4, so S is least.
7. 7 · Find the other values. h = 32/16 = 2 m. Least area S = 16 + 128/4 = 16 + 32 = 48 m².
8. 8 · Check and box the answer. Base 4 m by 4 m, height 2 m, least sheet metal 48 m². Check: 4 × 4 × 2 = 32 m³.

Scoring tip: Write h in terms of x before you differentiate. Show the second derivative test: the numbers alone get only part marks.