Optimisation problemsEdexcel International A Level Maths: Subtopic test

1. 1
   A farmer uses 80 m of fencing to enclose a rectangular area against a straight wall. The wall forms one side, so fencing is needed on only three sides. The two sides perpendicular to the wall each have length $x$ metres, and the enclosed area is $A$ m$^2$.

   (a)

   Find an expression for $A$ in terms of $x$.

   [1 mark]

   • A$80x-x^2$
   • B$80x-2x^2$
   • C$160x-2x^2$
   • D$40x-x^2$

   (b)

   Find the value of $x$ that gives the maximum area.

   [1 mark]

   • A$x=10$
   • B$x=40$
   • C$x=20$
   • D$x=800$

   (c)

   Show that this value of $x$ gives a maximum, and find the maximum area.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   An open-topped box has a square base of side $x$ cm and height $h$ cm, and its volume is $500$ cm$^3$. Its total outside surface area (the base and four sides) is $S$ cm$^2$.

   (a)

   Find an expression for $S$ in terms of $x$.

   [1 mark]

   • A$2x^2+\frac{2000}{x}$
   • B$x^2+\frac{500}{x}$
   • C$x^2+\frac{4x^3}{500}$
   • D$x^2+\frac{2000}{x}$

   (b)

   Find the value of $x$ for which $S$ is stationary.

   [1 mark]

   • A$x=10$
   • B$x=100$
   • C$x=\sqrt[3]{500}\approx7.94$
   • D$x=\sqrt{1000}\approx31.6$

   (c)

   Given that $S$ has a minimum value at this stationary point, find the minimum value of $S$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A company sells $x$ hundred phone cases each day. Its daily profit, $P$ hundred pounds, is modelled by $P=-x^3+9x^2-15x-10$, for $x\ge0$.

   (a)

   Show that $P$ is stationary when $x=1$ and when $x=5$.

   [3 marks]

   (b)

   Determine the number of phone cases that gives the maximum daily profit, justifying that it is a maximum, and find that maximum profit.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A closed cylinder has radius $r$ cm and height $h$ cm. Its volume is $432\pi$ cm$^3$ and its total surface area is $S$ cm$^2$.

   (a)

   (i) Show that $S=2\pi r^2+\frac{864\pi}{r}$.
   (ii) Find the value of $r$ for which $S$ is stationary.

   [6 marks]

   (b)

   (i) Show that this value of $r$ gives a minimum value of $S$.
   (ii) Find the minimum value of $S$ in the form $k\pi$.
   (iii) Find the corresponding value of $h$ and describe how it is related to $r$.

   [6 marks]

   Total for question 4: 12 marks

End of questions