A piece of wire 80\(~\text{cm}\) long is bent to form a rectangle. The dimensions to maximise the area of the rectangle are?

A rectangular garden is to have an area of 64\(~\text{m}^2\). What are the dimensions of the garden if the perimeter is to be a minimum?

A closed cylinder container is to have a volume of \(600\pi~\text{cm}^3\). The surface area \(S\), can be expressed as?

A box is made from an 8\(~\text{cm}\) by 4\(~\text{cm}\) rectangle or cardboard by cutting out 4 equal squares of side \(x~\text{cm}\)  from each corner. The edges are turned up to form a box. The volume of the box \(V\) is given by the formula?

A rectangular prism has a base with length twice its breadth. The volume is \(400~\text{cm}^3\).Given that the breadth is \(x~\text{cm}\), the surface area \(S\) is given by :

An open rectangular box has four sides and a base, but no lid, as in the figure.

The dimensions of the base are: length \(3x \) cm, width \(x \) cm and height \(y\) cm

(i)  Given that the surface area of the box is \(120c{m^2}\) show that \[y = \frac{{120 - 3{x^2}}}{{8x}}\]

(ii)  Show that the volume of the box is \[V = 9x\left( {5 - \frac{{{x^2}}}{8}} \right)\]

(iii)  Hence determine the maximum volume of the box 

A book is designed so that each page of print contains \(216\,c{m^2}\), surrounded completely by borders as illustrated in the figure. Each page is to have a border of \(1\,cm\) at the bottom and each side as well as a border of \(2\,cm\) at the top. Let the width of a page be \(x\,cm\) and its length \(y\,cm\)

(i)  Show that the area \(A\,c{m^2}\) of one page is \[A = x\left( {3 + \frac{{216}}{{x - 2}}} \right)\]

(ii)  Prove that the smallest print area possible will have dimensions \(12\,cm\) wide and \(18\,cm\) long

A farmer wishes to construct two rectangular enclosures, as shown above. Pen 2 is to be 5 times as wide as Pen1. There is an existing wall (shaded) that serves as a boundary fence as shown. All the other fences are to be constructed from 48 meters of wire mesh.

(i)  Let \(x\) be the length of both pens and y the width of Pen 1. Show that \(y = 8 - \frac{x}{2}\)

(ii)  Hence show that the total area \(A\,{m^2}\) contained in the two enclosures is given by \(A = 6x\left( {8 - \dfrac{x}{2}} \right)\)

(iii)   Calculate the maximum area of each pen

The figure shown above consists of a rectangle, \(x\) cm by \(y\) cm with two equilateral triangles on each side of length \(x\) cm. Given that the perimeter of the figure is 26 cm,

(i) Show that \(\displaystyle A = 13x - \frac{(4-\sqrt{3})x^2}{2}\) where \(A\) is the area of the figure.
(ii) Hence show that the maximum area of the figure is \(\displaystyle \frac{13(4+\sqrt{3})}{2} \text{ cm}^2\)

The diagram shows a straight section of river, 3 km wide. Tom is at the point \(O\) on one bank and he wishes to reach point \(B\) on the opposite bank.

The point \(A\) is directly opposite \(O\) and the distance from \(A\) to \(B\) is 6 km.

Tom can row at 8 km/hour and bicycle at 10 km/hour. He intends to row in a straight line to a point \(P\) on the opposite bank and then bicycle directly from \(P\) to \(B\). Let the distance \(AP\) be \(x\) cm.

(i) Show that the time \(T\), in hours, that Tom takes to reach \(B\) is given by \( \displaystyle T = \frac{\sqrt{x^2 + 9}}{8} + \frac{6-x}{10}\)
(ii) Find how far from his destination Tom must land on the opposite bank in order to reach \(B\) in the shortest time possible.

Bill and Ben set out for a town. Bill is 6 km West of the town and walking at a constant 3km/hr. Ben is 4km South of the town and walking at a constant rate of 4km/hr.

(i)  Show that their distance apart after t hours is given by \({D^2} = 25{t^2} - 68t + 52\)

(ii)  Hence find how long it takes them to reach their minimum distance apart

(iii)  Find their minimum distance apart

The diagram shows the graph of the derivative of a certain function \(f(x)\). Copy this graph and on the same set of axes, draw a sketch of \(f(x)\)

The diagram shows a sector of a circle with radius \(r\), the angle at the centre is \(\theta\) radians and the area of the sector is \(8\text{ cm}^2\).

(i) Find an expression for \(r\) in terms of \(\theta\).

(ii) Show that the perimeter of the sector is given by \(P = \dfrac{8}{\sqrt{\theta}} + 4\sqrt{\theta}\).

(iii) If \(0 < \theta \leq \pi\), find the value of \(\theta\) for a minimum perimeter.

The cost per hour of a bike ride is given by the formula \(C = {x^2} - 12x + 60\), where \(x\) is the distance traveled in km. The distance that gives the minimum cost is ? 

The surface area of a cylinder is given by \(S = 2\pi {r^2} + \dfrac{{500\pi }}{r}\) . The value of \(r\) that will give a minimum surface area is?

A 2\(m\) length of wire is cut into 2 pieces and each piece is bent to form a square. Given that the length of a piece is \(x m\), then the total area \(A\) of the two squares can be expressed as ?

A book is designed so that each page of print contains \(216cm^2\), surrounded completely by borders as illustrated in the figure. Each page is to have a border of 1cm at the bottom and at each side, as well as a border of 2cm at the top. Let the width of the page be \(x\) and its length \(y\)cm.

(i) Show that the area \(A\) \(cm^2\) of one page is \(A = x\left( {3 + \dfrac{{216}}{{x - 2}}} \right)\) (3 marks)

(ii) Prove that the smallest print area possible will have dimensions 12cm wide and 18cm long. (4 marks)

Nathan has designed a garden bed which consists of a rectangle of width \(y\) metres and length \(4 x\) metres, and a semi-circle as shown in the diagram, given that the perimeter is 50m:

i) Show that the perimeter of this garden bed can be expressed as \(2 x \pi+2 y+4 x=50\)

ii) Rearrange the above perimeter to express \(y\) in terms of \(x\).

iii) Show that the area of the garden bed can be given by the formula \(A=100 x-8 x^2-2 x^2 \pi\)

iv) Find the value of \(x\) that gives the maximum area. Correct your answer to 2 decimal places.