A-LevelMathematicsDifferentiation9709/12 · Oct/Nov 2010 · Q1010 Marks
The diagram shows an open rectangular tank of height h metres covered with a lid. The base of the tank has sides of length x metres and 5/4x metres and the lid is a rectangle with sides of length x metres and 5/4x metres. When full the tank holds 4 m³ of water. The material from which the tank is made is of negligible thickness. The external surface area of the tank together with the area of the top of the lid is A m².
- (i)Express h in terms of x and hence show that A = (3/2)x² + 24/x.[5]
- (ii)Given that x can vary, find the value of x for which A is a minimum, showing clearly that A is a minimum and not a maximum.[5]
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Mark scheme
Marking points from the official mark scheme (Oct/Nov 2010). The mark or mark type is shown in brackets.
- [M1] h=8/x² Uses lbh = 4
- [A1] A = x*x + 5/4x*x + 2xh + 2xh + 5/4x*h + 5/4x*h co
- [M1] A = (3/2)x² + 3xh Allow 1 error but needs the lid
- [M1] A = (3/2)x² + 3x * (8/x²) For substitution of h as f(x)
- [A1] A = (3/2)x² + 24/x AG
- [B1] dA/dx = 3x - 24/x² = 0 Correct derivative.
- [M1] Sets to 0 and attempts to solve.
- [A1] x = 2 co
- [M1] d²A/dx² = 3 + 48/x³ Reasonable attempt – allow 1 error
- [A1] > 0 when x = 2 hence minimum co AG (Result consistent with their f'')
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