5.7 OptimisationIB Maths: Applications and Interpretation SL: Subtopic test

1. 1
   A rectangular garden has a perimeter of 60 m. Its width is $x$ metres.

   (a)

   Find an expression for the area $A$ of the garden in terms of $x$.

   [1 mark]

   • A$x(60-x)$
   • B$x(60-2x)$
   • C$x(30+x)$
   • D$x(30-x)$

   (b)

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

   [1 mark]

   • A$x=15$
   • B$x=30$
   • C$x=7.5$
   • D$x=60$

   (c)

   Find the maximum area of the garden.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A factory produces $x$ tonnes of steel each day. The daily cost, $C$ thousand USD, is modelled by $C(x)=2x^2-48x+500$.

   (a)

   Find $C'(x)$.

   [1 mark]

   • A$2x-48$
   • B$4x^2-48x$
   • C$4x-48$
   • D$4x+48$

   (b)

   Find the number of tonnes per day that gives the minimum daily cost, ignoring any restriction on $x$.

   [1 mark]

   • A$x=48$
   • B$x=12$
   • C$x=24$
   • D$x=-12$

   (c)

   The factory has a contract that requires it to produce between 15 and 30 tonnes per day. Find the minimum daily cost under this contract, in USD.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A square sheet of card has side 30 cm. A square of side $x$ cm is cut from each corner and the sides are folded up to make an open box with no lid, where $0<x<15$. The volume of the box is $V$ cm³.

   (a)

   Show that $V=4x^3-120x^2+900x$.

   [3 marks]

   (b)

   Use your GDC to find the value of $x$ that gives the maximum volume, and find this maximum volume.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A closed cylinder has radius $r$ cm, height $h$ cm and total surface area 600 cm².

   (a)

   (i) Show that the volume of the cylinder is $V=300r-\pi r^3$.
   (ii) Find $\frac{dV}{dr}$ and hence find the value of $r$ for which $V$ is stationary.

   [6 marks]

   (b)

   (i) Justify that the value of $r$ found in part (a) gives a maximum volume.
   (ii) Find the maximum volume of the cylinder.
   (iii) Find the height of the cylinder for this maximum volume.

   [6 marks]

   Total for question 4: 12 marks

End of questions