Edexcel IALMathematicsGeneral Applications10 Marks
A rectangle has width x cm and length y cm. The perimeter of the rectangle is 20 cm.
- (a)Show that the area of the rectangle, A cm^2, is given by A = 10x - x^2.[2]
- (b)Use calculus to find the maximum area of the rectangle.[5]
- (c)State the dimensions of the rectangle that give this maximum area.[3]
Mark scheme
Marking points from the official mark scheme. The mark or mark type is shown in brackets.
- M1: Uses 2x + 2y = 20 to express y in terms of x (or vice versa). E.g. y = 10 - x.
- A1*: Substitutes y into A = xy to show A = x(10 - x) = 10x - x^2.
- M1: Differentiates A with respect to x (at least one term correct).
- A1: dA/dx = 10 - 2x.
- M1: Sets dA/dx = 0 and solves for x, getting x = 5.
- M1: Finds the second derivative d^2A/dx^2 = -2 and states it's less than 0, or uses another valid method to justify it's a maximum (e.g. considering the shape of the quadratic).
- A1: Substitutes x=5 into A = 10x - x^2 to find max area = 25.
- B1ft: States x=5 (from part b).
- M1: Finds y using x=5: y = 10 - 5 = 5.
- A1: States dimensions are 5cm by 5cm (or square of side 5cm).
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