O-LevelAdditional MathematicsDifferentiation and integration4037/22 · May/June 2021 · Q118 Marks
In this question all lengths are in centimetres. The volume and surface area of a sphere of radius r are (4/3)πr³ and 4πr² respectively. [Figure 11.1] The diagram shows a solid object made from a hemisphere of radius x and a cylinder of radius x and height y. The volume of the object is 500 cm³.
- (a)Find an expression for y in terms of x and show that the surface area, S, of the object is given by S = (5/3)πx² + 1000/x.[4]
- (b)Given that x can vary and that S has a minimum value, find the value of x for which S is a minimum.[4]
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Mark scheme
Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.
- (a) M1 500 = (4/6)πx³ + πx²y oe
- (a) A1 if first M0, SC1 for y = (1/(πx²))(500 - (4/3)πx³) oe, isw
- (a) M1 dep on first M1 S = 2πx² + πx² + 2πx((500/(πx²)) - (2/3)x) Correct completion to given answer: S = (5/3)πx² + 1000/x
- (a) A1 Correct completion to given answer: S = (5/3)πx² + 1000/x
- (b) B2 B1 for each term Differentiates S: (10/3)πx - 1000/x² oe
- (b) M1 FT their dS/dx = 0 providing at least B1 awarded (10/3)πx - 1000/x² = 0 and attempt to solve
- (b) A1 x = ³√(300/π) isw or 4.57[07...] nfww
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