Mathematical problem solvingAQA A-Level Maths: Subtopic test

1. 1
   A rectangular garden is fenced on three sides using 40 m of fencing, the fourth side being a wall. Each of the two sides perpendicular to the wall has length $x$ metres, and the area of the garden is $A\ \mathrm{m}^2$.

   (a)

   Which expression gives $A$ in terms of $x$?

   [1 mark]

   • A$40x-2x^2$
   • B$40x-x^2$
   • C$20x-x^2$
   • D$2x(40-x)$

   (b)

   What is the maximum possible area of the garden?

   [1 mark]

   • A$400\ \mathrm{m}^2$
   • B$100\ \mathrm{m}^2$
   • C$200\ \mathrm{m}^2$
   • D$20\ \mathrm{m}^2$

   (c)

   State the values of $x$ for which the model is realistic, justifying your answer.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Firm A charges a fixed £3.50 plus £1.80 per mile for a taxi journey. Firm B charges a fixed £2.00 plus £2.20 per mile.

   (a)

   What does a 5-mile journey cost with Firm A?

   [1 mark]

   • A£9.00
   • B£12.50
   • C£5.30
   • D£17.50

   (b)

   For what journey length do the two firms charge the same?

   [1 mark]

   • A$1.5$ miles
   • B$2.5$ miles
   • C$3.0$ miles
   • D$3.75$ miles

   (c)

   A customer is making a 6-mile journey. Which firm should they choose, and by how much does it save them?

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A cylindrical water tank has internal radius 1.5 m and internal height 2.0 m.

   (a)

   Find the capacity of the tank in litres, to 3 significant figures. (1 m$^3$ = 1000 litres.)

   [3 marks]

   (b)

   The tank is filled from empty by a pump delivering 25 litres per minute, but it leaks at 5 litres per minute. Find the time taken to fill it, to the nearest minute, and state one assumption and its likely effect.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A farmer has 120 m of fencing to build a rectangular enclosure of length $L$ m and width $w$ m. It is divided into three equal pens by two internal fences, each parallel to a side of length $w$.

   (a)

   (i) Show that $L=60-2w$. (ii) Hence show that the total area is $A=60w-2w^2$ and find the width that gives the largest area, and that area.

   [6 marks]

   (b)

   The farmer needs each pen to have an area of at least 120 m$^2$. (i) Find the possible values of $w$, giving the bounds to 3 s.f. (ii) Check that these values are valid in context. (iii) State one limitation of the model.

   [6 marks]

   Total for question 4: 12 marks

End of questions