A metal tank is in the shape of a cuboid with a square base of side xm and an open top. The tank has a volume of 5 m³. Given that x can vary, and that the area of the metal used to make the tank is a minimum, find the dimensions of the tank.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

1. [M2] Correct use of x²h = 5 to find an expression that can be used to eliminate h [M1] for x²h = 5 soi
2. [B1] Surface area: x² + 4x (5/x²) oe [FT] their surface area of form ax² + b/x
3. [B1] Derivative of the surface area: 2x - 20x⁻² oe [FT] their derivative of form ax + b/x²
4. [M1] Equates their 2x – 20x⁻² to 0 and solves to find a value of x
5. [A1] x = 2.15 or 2.154[...] h = 1.08 or 1.077[...] nfww [dep] on all previous marks awarded
6. Alternative
7. [(M2)] Correct use of x²h = 5 to find an expression that can be used to eliminate x [M1] for x²h = 5 soi
8. [(B1)] Surface area: (√5/h)² + 4h√(5/h) oe
9. [(B1)] Derivative of the surface area: 4√5 x h^(-1/2) - 5/h² oe [FT] their surface area of form a/h + b√h
10. [(M1)] Equates their 4√5 x h^(-1/2) - 5/h² to 0 and solves to find a value of h
11. [(A1)] h = 1.08 or 1.077[...] x = 2.15 or 2.154[...] nfww [dep] on all previous marks awarded