A cylinder has radius \(r\text{ cm}\) and height \(h\text{ cm}\). The total surface area, including the two ends, is \(A\text{ cm}^2\). The volume of the cylinder is \(330\text{ cm}^3\).

(a) Show that \(A=2\pi r^2+\dfrac{660}{r}\).

(b) Given that \(r\) can vary, find the value of \(r\) that gives a stationary value for \(A\) and show that this value is a minimum.

A metal tank is in the shape of a cuboid with a square base of side \(x \mathrm{~m}\) and an open top. The tank has a volume of \(5 \mathrm{~m}^{3}\). Given that \(x\) can vary, and that the area of the metal used to make the tank is a minimum, find the dimensions of the tank.

In this question all lengths are in centimetres.

The diagram shows a rectangle \(A B C D\) with \(B C=x\). The area of the rectangle is \(400 \mathrm{~cm}^{2}\). Two identical quarter-circles of radius \(\frac{x}{2}\), with centres \(A\) and \(C\), are removed from the rectangle to make the shaded shape.

Given that \(x\) can vary, find the value of \(x\) that gives the minimum value of the perimeter of the shaded shape and hence find this minimum value.

The curved surface area of a cylinder with radius \(r\) and height \(h\) is \(2\pi rh\).

(a) A closed cylinder has volume \(1000\text{ cm}^3\). Show that its total surface area, \(S\text{ cm}^2\), is given by

\(S=2\pi r^2+\frac{2000}{r}\).

(b) Find the value of \(r\) for which the total surface area is a minimum.

In this question all lengths are in centimetres.

The diagram shows a right triangular prism of height \(h\) inside a right pyramid. The pyramid has a height of \(12\) and a base that is an equilateral triangle, \(ABC\), of side \(8\). The base of the prism sits on the base of the pyramid. Points \(P\), \(Q\) and \(R\) lie on the edges \(OA\), \(OB\) and \(OC\), respectively, of the pyramid \(OABC\). Pyramids \(OABC\) and \(OPQR\) are similar.

(a) Show that the volume, \(V\), of the triangular prism is given by

\(V=\frac{\sqrt3}{9}(ah^3+bh^2+ch),\)

where \(a\), \(b\) and \(c\) are integers to be found.

(b) It is given that, as \(h\) varies, \(V\) has a maximum value. Find the value of \(h\) that gives this maximum value of \(V\).

The diagram shows an open container in the shape of a half-cylinder. The length is \(y\) cm and the radius is \(x\) cm. The volume of the container is \(25000\text{ cm}^3\). Given that the outer surface area \(S\text{ cm}^2\) has a minimum value, find this minimum value.

In this question all lengths are in centimetres.

The volume of a cylinder with radius \(r\) and height \(h\) is \(\pi r^2h\) and its curved surface area is \(2\pi rh\).

The volume of a sphere with radius \(r\) is \(\frac43\pi r^3\) and its surface area is \(4\pi r^2\).

The diagram shows a solid object in the shape of a cylinder of base radius \(r\) and height \(h\), with a hemisphere of radius \(r\) on top. The total surface area of the object is \(300\text{ cm}^2\).

(a) Find an expression for \(h\) in terms of \(r\).

(b) Show that the volume, \(V\), of the object is

\(150r-\frac56\pi r^3.\)

(c) Find the maximum volume of the object as \(r\) varies.

In this question all lengths are in centimetres.

The volume and surface area of a sphere of radius \(r\) are \(\frac43\pi r^3\) and \(4\pi r^2\) respectively.

The diagram shows a solid object made from a hemisphere of radius \(x\) and a cylinder of radius \(x\) and height \(y\). The volume of the object is \(500\text{ cm}^3\).

(a) Find an expression for \(y\) in terms of \(x\) and show that the surface area, \(S\), of the object is given by

\(S=\frac53\pi x^2+\frac{1000}{x}.\)

(b) Given that \(x\) can vary and that \(S\) has a minimum value, find the value of \(x\) for which \(S\) is a minimum.

In this question all lengths are in centimetres.

The volume and curved surface area of a cone of base radius \(r\), height \(h\) and sloping edge \(l\) are \(\frac13\pi r^2h\) and \(\pi rl\) respectively.

The diagram shows a cone of base radius \(x\), height \(y\) and sloping edge \(\sqrt{x^2+y^2}\). The volume of the cone is \(10\pi\).

(a) Find an expression for \(y\) in terms of \(x\) and show that the curved surface area, \(S\), of the cone is given by

\(S=\frac{\pi\sqrt{x^6+900}}{x}.\)

(b) Given that \(x\) can vary and that \(S\) has a minimum value, find the exact value of \(x\) for which \(S\) is a minimum.

The volume \(V\) of a cone with base radius \(r\) and vertical height \(h\) is given by

\(V=\frac13\pi r^2h.\)

The curved surface area of a cone with base radius \(r\) and slant height \(l\) is given by \(\pi rl\).

A cone has base radius \(r\text{ cm}\), vertical height \(h\text{ cm}\) and volume \(V\text{ cm}^3\). The curved surface area of the cone is \(4\pi\text{ cm}^2\).

(a) Show that

\(h^2=\frac{16}{r^2}-r^2.\)

(b) Show that

\(V=\frac{\pi}{3}\sqrt{16r^2-r^6}.\)

(c) Given that \(r\) can vary and that \(V\) has a maximum value, find the value of \(r\) that gives the maximum volume.

An open cylinder has radius \(r\) cm and height \(h\) cm. Its volume is \(1000\text{ cm}^3\).

Find the minimum possible value of the total outer surface area of the cylinder.

The diagram shows a rectangular field \(ABDE\), where \(AB=300\) m and \(AE=400\) m. Joseph walks from \(A\) to \(C\) across the field at \(0.9\text{ m s}^{-1}\), then from \(C\) to \(D\) along the edge of the field at \(1.5\text{ m s}^{-1}\). It is given that \(BC=x\) metres.

(a) Show that the total time, \(T\) seconds, for Joseph's walk is

\(T=\frac{\sqrt{300^2+x^2}}{0.9}+\frac{400-x}{1.5}.\)

(b) Find the minimum possible value of \(T\).

A sector of a circle has radius \(r\) cm and area \(36\text{ cm}^2\). The perimeter of the sector is \(P\) cm.

(i) Show that \(P=2r+\dfrac{72}{r}\).

(ii) Find the stationary value of \(P\), and determine whether it is a maximum or a minimum.

A closed cylinder has base radius \(r\), height \(h\), and volume \(V\). The total surface area of the cylinder is \(600\pi\), and \(V\), \(r\) and \(h\) can vary.

(i) Show that \(V=300\pi r-\pi r^3\).

(ii) Find the stationary value of \(V\) and determine its nature.

The diagram shows an open container in the shape of a cuboid of width \(x\) cm, length \(4x\) cm and height \(h\) cm. The volume of the container is \(800\text{ cm}^3\).

(i) Show that the external surface area, \(S\text{ cm}^2\), of the open container is such that \(S=4x^2+\frac{2000}{x}\).

(ii) Given that \(x\) can vary, find the stationary value of \(S\) and determine its nature.

A solid circular cylinder has a base radius of \(r\) cm and a height of \(h\) cm. The cylinder has a volume of \(1200\pi\text{ cm}^3\) and a total surface area of \(S\text{ cm}^2\).

(i) Show that

\(S=2\pi r^2+\frac{2400\pi}{r}.\)

(ii) Given that \(h\) and \(r\) can vary, find the stationary value of \(S\) and determine its nature.

In this question, all lengths are in metres.

The diagram shows a window formed by a semi-circle of radius \(r\) on top of a rectangle with dimensions \(2r\) by \(y\). The total perimeter of the window is \(5\).

(i) Find \(y\) in terms of \(r\).

(ii) Show that the total area of the window is

\(A=5r-\frac{\pi r^2}{2}-2r^2.\)

(iii) Given that \(r\) can vary, find the value of \(r\) which gives a maximum area of the window and find this area. You are not required to show that this area is a maximum.

The volume of a closed cylinder of base radius \(x\text{ cm}\) and height \(h\text{ cm}\) is \(500\text{ cm}^3\).

(i) Express \(h\) in terms of \(x\).

(ii) Show that the total surface area of the cylinder is given by \(A=2\pi x^2+\dfrac{1000}{x}\text{ cm}^2\).

(iii) Given that \(x\) can vary, find the stationary value of \(A\) and show that this value is a minimum.