A farmer owns a field in the shape of a triangle $\text{ABC}$ such that $\text{AB}=650 \text{m}, \text{AC}=1005 \text{m}$ and $\text{BC}=1225 \text{m}$.

The local town is planning to build a highway that will intersect the borders of the field at points $\text{D}$ and $\text{E}$, where $\text{DC}=210 \text{m}$ and $\text{CÊD}=100°$, as shown in the diagram below.

The town wishes to build a carpark here. They ask the farmer to exchange the part of the field represented by triangle $\text{DCE}$. In return the farmer will get a triangle of equal area $\text{ADF}$, where $\text{F}$ lies on the same line as $\text{D}$ and $\text{E}$, as shown in the diagram above.

Using geometry software, Pedro draws a quadrilateral $\text{ABCD}$. $\text{AB}=8 \text{cm}$ and $\text{CD}=9 \text{cm}$. Angle $\text{BAD}=51.5°$ and angle $\text{ADB}=52.5°$. This information is shown in the diagram.

$\text{CE}=7 \text{cm}$, where point $\text{E}$ is the midpoint of $\text{BD}$.

The living accommodation on a university campus is in the shape of a rectangle with sides of length $200 \text{m}$ and $300 \text{m}$.

There are three offices for the management of the accommodation set at the points $\text{A}$, $\text{B}$ and $\text{C}$. These offices are responsible for all the students in the areas closest to the office. These areas are shown on the Voronoi diagram below. On this coordinate system the positions of $\text{A}$, $\text{B}$ and $\text{C}$ are $(100, 160)$, $(100, 40)$ and $(250, 100)$ respectively.

The equation of the perpendicular bisector of $\left[\text{AC}\right]$ is $5x-2y=615$.

The manager of office $\text{C}$ believes that he has more than one third of the area of the campus to manage.

The following diagram shows a circle with centre $\text{O}$ and radius $1 \text{cm}$. Points $\text{A}$ and $\text{B}$ lie on the circumference of the circle and $\text{A}\hat{\text{O}}\text{B}=2\theta$, where $0<\theta <\frac{\mathrm{\pi }}{2}$.

The tangents to the circle at $\text{A}$ and $\text{B}$ intersect at point $\text{C}$.

The diagram below shows a circular clockface with centre $\text{O}$. The clock’s minute hand has a length of $10 \text{cm}$. The clock’s hour hand has a length of $6 \text{cm}$.

At $4:00$ pm the endpoint of the minute hand is at point $\text{A}$ and the endpoint of the hour hand is at point $\text{B}$.

Between $4:00$ pm and $4:13$ pm, the endpoint of the minute hand rotates through an angle, $\theta$, from point $\text{A}$ to point $\text{C}$. This is illustrated in the diagram.

A second clock is illustrated in the diagram below. The clock face has radius $10 \text{cm}$ with minute and hour hands both of length $10 \text{cm}$. The time shown is $6:00$ am. The bottom of the clock face is located $3 \text{cm}$ above a horizontal bookshelf.

The height, $h$ centimetres, of the endpoint of the minute hand above the bookshelf is modelled by the function

$h\left(\theta \right)=10 \cos \theta +13, \theta \ge 0,$

where $\theta$ is the angle rotated by the minute hand from $6:00$ am.

The height, $g$ centimetres, of the endpoint of the hour hand above the bookshelf is modelled by the function

$g\left(\theta \right)=-10 \cos \left(\frac{\theta }{12}\right)+13, \theta \ge 0,$

where $\theta$ is the angle in degrees rotated by the minute hand from $6:00$ am.

A hollow chocolate box is manufactured in the form of a right prism with a regular hexagonal base. The height of the prism is $h \text{cm}$, and the top and base of the prism have sides of length $x \text{cm}$.

A large water reservoir is built in the form of part of an upside-down right pyramid with a horizontal square base of length $80$ metres. The point $\text{C}$ is the centre of the square base and point $\text{V}$ is the vertex of the pyramid.

The bottom of the reservoir is a square of length $60$ metres that is parallel to the base of the pyramid, such that the depth of the reservoir is $6$ metres as shown in the diagram.

The second diagram shows a vertical cross section, $\text{MNOPC}$, of the reservoir.

Every day $80 {\text{m}}^3$ of water from the reservoir is used for irrigation.

Joshua states that, if no other water enters or leaves the reservoir, then when it is full there is enough irrigation water for at least one year.

The Tower of Pisa is well known worldwide for how it leans.

Giovanni visits the Tower and wants to investigate how much it is leaning. He draws a diagram showing a non-right triangle, ABC.

On Giovanni’s diagram the length of AB is 56 m, the length of BC is 37 m, and angle ACB is 60°. AX is the perpendicular height from A to BC.

Giovanni’s tourist guidebook says that the actual horizontal displacement of the Tower, BX, is 3.9 metres.

A sector of a circle, centre $\text{O}$ and radius $4.5 \text{m}$, is shown in the following diagram.

A square field with side $8 \text{m}$ has a goat tied to a post in the centre by a rope such that the goat can reach all parts of the field up to $4.5 \text{m}$ from the post.

^{[Source: mynamepong, n.d. Goat [image online] Available at: https://thenounproject.com/term/goat/1761571/
This file is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0)
https://creativecommons.org/licenses/by-sa/3.0/deed.en [Accessed 22 April 2010] Source adapted.]}

Let $V$ be the volume of grass eaten by the goat, in cubic metres, and $t$ be the length of time, in hours, that the goat has been in the field.

The goat eats grass at the rate of $\frac{dV}{dt}=0.3 t{\text{e}}^{-t}$.

A king rules a small mountain kingdom which is in the form of a square of length 4 kilometres. The square is described by the co-ordinate system $0⩽<!--\; ⩽\; -->x⩽<!--\; ⩽\; -->4\text{,}0⩽<!--\; ⩽\; -->y⩽<!--\; ⩽\; -->4$.

The king has four adult children, each of which has a luxury palace located at the points $\left(1\text{,}1\right)\text{,}\left(3\text{,}1\right)\text{,}\left(1\text{,}3\right)\text{,}\left(3\text{,}3\right)$. Each child owns all the land that is nearer their palace than any other palace.

The king has a fifth (youngest) child who is now just growing up. He installs her in a new palace situated at point (2, 2). The rule about ownership of land is then reapplied.

The Voronoi diagram below shows four supermarkets represented by points with coordinates $\text{A}(0, 0)$, $\text{B}(6, 0)$, $\text{C}(0, 6)$ and $\text{D}(2, 2)$. The vertices $\text{X}$, $\text{Y}$, $\text{Z}$ are also shown. All distances are measured in kilometres.

The equation of $\text{(XY)}$ is $y=2-x$ and the equation of $\text{(YZ)}$ is $y=0.5x+3.5$.

The coordinates of $\text{Y}$ are $(-1, 3)$ and the coordinates of $\text{Z}$ are $(7, 7)$.

A town planner believes that the larger the area of the Voronoi cell $\text{XYZ}$, the more people will shop at supermarket $\text{D}$.

A pan, in which to cook a pizza, is in the shape of a cylinder. The pan has a diameter of 35 cm and a height of 0.5 cm.

A chef had enough pizza dough to exactly fill the pan. The dough was in the shape of a sphere.

The pizza was cooked in a hot oven. Once taken out of the oven, the pizza was placed in a dining room.

The temperature, $P$, of the pizza, in degrees Celsius, °C, can be modelled by

$$P(t)=a(2.06{)}^{-<!--\; -\; -->t}+19,t⩾<!--\; ⩾\; -->0$$

where $a$ is a constant and $t$ is the time, in minutes, since the pizza was taken out of the oven.

When the pizza was taken out of the oven its temperature was 230 °C.

The pizza can be eaten once its temperature drops to 45 °C.

Abdallah owns a plot of land, near the river Nile, in the form of a quadrilateral ABCD.

The lengths of the sides are $\text{AB}=\text{40 m, BC}=\text{115 m, CD}=\text{60 m, AD}=\text{84 m}$ and angle $\mathrm{B}\stackrel{^<!--\; ^\; -->}{\mathrm{A}}\mathrm{D}={90}^{\circ <!--\; \circ \; -->}$.

This information is shown on the diagram.

The formula that the ancient Egyptians used to estimate the area of a quadrilateral ABCD is

$\text{area}=\frac{(\text{AB}+\text{CD})(\text{AD}+\text{BC})}{4}$.

Abdallah uses this formula to estimate the area of his plot of land.

A water container is made in the shape of a cylinder with internal height $h$ cm and internal base radius $r$ cm.

The water container has no top. The inner surfaces of the container are to be coated with a water-resistant material.

The volume of the water container is $0.5{\text{m}}^3$.

The water container is designed so that the area to be coated is minimized.

One can of water-resistant material coats a surface area of $2000\text{ c}{\text{m}}^2$.

A factory packages coconut water in cone-shaped containers with a base radius of 5.2 cm and a height of 13 cm.

The factory designers are currently investigating whether a cone-shaped container can be replaced with a cylinder-shaped container with the same radius and the same total surface area.

An archaeological site is to be made accessible for viewing by the public. To do this, archaeologists built two straight paths from point A to point B and from point B to point C as shown in the following diagram. The length of path AB is 185 m, the length of path BC is 250 m, and angle $\text{A}\stackrel{\wedge <!--\; \wedge \; -->}{\text{B}}<!--\; \; -->\text{C}$ is 125°.

The archaeologists plan to build two more straight paths, AD and DC. For the paths to go around the site, angle $\text{B}\stackrel{\wedge <!--\; \wedge \; -->}{\text{A}}<!--\; \; -->\text{D}$ is to be made equal to 85° and angle $\text{B}\stackrel{\wedge <!--\; \wedge \; -->}{\text{C}}<!--\; \; -->\text{D}$ is to be made equal to 70° as shown in the following diagram.

The diagram shows the straight line $L_1$. Points $\text{A}\left(-9, -1\right)$, $\text{M}\left(-3, 2\right)$ and $\text{C}$ are points on $L_1$.

$\text{M}$ is the midpoint of $\text{AC}$.

Line $L_2$ is perpendicular to $L_1$ and passes through point $\text{M}$.

The point $\text{N}\left(k, 4\right)$ is on $L_2$.

Farmer Brown has built a new barn, on horizontal ground, on his farm. The barn has a cuboid base and a triangular prism roof, as shown in the diagram.

The cuboid has a width of 10 m, a length of 16 m and a height of 5 m.
The roof has two sloping faces and two vertical and identical sides, ADE and GLF.
The face DEFL slopes at an angle of 15° to the horizontal and ED = 7 m .

The roof was built using metal supports. Each support is made from five lengths of metal AE, ED, AD, EM and MN, and the design is shown in the following diagram.

ED = 7 m , AD = 10 m and angle ADE = 15° .
M is the midpoint of AD.
N is the point on ED such that MN is at right angles to ED.

Farmer Brown believes that N is the midpoint of ED.

A manufacturer makes trash cans in the form of a cylinder with a hemispherical top. The trash can has a height of 70 cm. The base radius of both the cylinder and the hemispherical top is 20 cm.

A designer is asked to produce a new trash can.

The new trash can will also be in the form of a cylinder with a hemispherical top.

This trash can will have a height of H cm and a base radius of r cm.

There is a design constraint such that H + 2r = 110 cm.

The designer has to maximize the volume of the trash can.

A restaurant serves desserts in glasses in the shape of a cone and in the shape of a hemisphere. The diameter of a cone shaped glass is 7.2 cm and the height of the cone is 11.8 cm as shown.

The volume of a hemisphere shaped glass is $225\text{ c}{\text{m}}^3$.

The restaurant offers two types of dessert.

The regular dessert is a hemisphere shaped glass completely filled with chocolate mousse. The cost, to the restaurant, of the chocolate mousse for one regular dessert is $1.89.

The special dessert is a cone shaped glass filled with two ingredients. It is first filled with orange paste to half of its height and then with chocolate mousse for the remaining volume.

The cost, to the restaurant, of $100\text{ c}{\text{m}}^3$ of orange paste is $7.42.

A chef at the restaurant prepares 50 desserts; $x$ regular desserts and $y$ special desserts. The cost of the ingredients for the 50 desserts is $111.44.

Let $f\left(x\right)=\frac{16}{x}$. The line $L$ is tangent to the graph of $f$ at $x=8$.

$L$ can be expressed in the form r $=\left(\begin{array}{c}8\\ 2\end{array}\right)+t$u.

The direction vector of $y=x$ is $\left(\begin{array}{c}1\\ 1\end{array}\right)$.

A farmer owns a plot of land in the shape of a quadrilateral ABCD.

$\text{AB}=105\text{ m, BC}=95\text{ m, CD}=40\text{ m, DA}=70\text{ m}$ and angle $\text{DCB}={90}^{\circ <!--\; \circ \; -->}$.

The farmer wants to divide the land into two equal areas. He builds a fence in a straight line from point B to point P on AD, so that the area of PAB is equal to the area of PBCD.

Calculate

The base of an electric iron can be modelled as a pentagon ABCDE, where:

$$\begin{array}{l}\text{BCDE is a rectangle with sides of length }(x+3)\text{ cm and }(x+5)\text{ cm;}\\ \text{ABE is an isosceles triangle, with AB}=\text{AE and a height of }x\text{ cm;}\\ \text{the area of ABCDE is 222 c}{\text{m}}^{\text{2}}\text{.}\end{array}$$

Insulation tape is wrapped around the perimeter of the base of the iron, ABCDE.

F is the point on AB such that $\text{BF}=\text{8 cm}$. A heating element in the iron runs in a straight line, from C to F.

The Maxwell Ohm Company is designing a portable Bluetooth speaker. The speaker is in the shape of a cylinder with a hemisphere at each end of the cylinder.

The dimensions of the speaker, in centimetres, are illustrated in the following diagram where $r$ is the radius of the hemisphere, and $l$ is the length of the cylinder, with $r>0$ and $l\ge 0$.

The Maxwell Ohm Company has decided that the speaker will have a surface area of $300 {\text{cm}}^2$.

The quality of sound from the speaker will improve as $V$ increases.

Haraya owns two triangular plots of land, $\text{ABC}$ and $\text{ACD}$. The length of $\text{AB}$ is $30 \text{m}$, $\text{BC}$ is $50 \text{m}$ and $\text{AC}$ is $70 \text{m}$. The size of $\text{D}\hat{\text{A}}\text{C}$ is $55°$ and $\text{A}\hat{\text{D}}\text{C}$ is $72°$.

The following diagram shows this information.

Haraya attaches a $20 \text{m}$ long rope to a vertical pole at point $\text{B}$.

Consider the points A(−3, 4, 2) and B(8, −1, 5).

A line L has vector equation $r=\left(\begin{array}{c}2\\ 0\\ -<!--\; -\; -->5\end{array}\right)+t\left(\begin{array}{c}1\\ -<!--\; -\; -->2\\ 2\end{array}\right)$. The point C (5, $y$, 1) lies on line L.

A communication tower, T, produces a signal that can reach cellular phones within a radius of 32 km. A straight road passes through the area covered by the tower’s signal.

The following diagram shows a line representing the road and a circle representing the area covered by the tower’s signal. Point R is on the circumference of the circle and points S and R are on the road. Point S is 38 km from the tower and RŜT = 43˚.

Two points P and Q have coordinates (3, 2, 5) and (7, 4, 9) respectively.

Let $\stackrel{\to <!--\; \to \; -->}{\text{PR}}$ = 6i − j + 3k.

The following diagram shows quadrilateral ABCD.

$\text{AB}=11\text{cm,}\text{BC}=6\text{cm,}\text{B}\stackrel{\wedge <!--\; \wedge \; -->}{\text{A}}<!--\; \; -->{\text{D = 100}}^{\circ <!--\; \circ \; -->}\text{, and C}\stackrel{\wedge <!--\; \wedge \; -->}{\text{B}}<!--\; \; -->{\text{D = 82}}^{\circ <!--\; \circ \; -->}$

Let $\stackrel{\to <!--\; \to \; -->}{\text{AB}}=\left(\begin{array}{c}4\\ 1\\ 2\end{array}\right)$.

The following diagram shows a circle, centre O and radius $r$ mm. The circle is divided into five equal sectors.

One sector is OAB, and $\mathrm{A}\stackrel{^<!--\; ^\; -->}{\mathrm{O}}\mathrm{B}=\mathrm{\theta <!--\; \theta \; -->}$.

The area of sector AOB is $20\mathrm{\pi <!--\; \pi \; -->}\text{ m}{\text{m}}^2$.

OAB is a sector of the circle with centre O and radius $r$, as shown in the following diagram.

The angle AOB is $\mathrm{\theta <!--\; \theta \; -->}$ radians, where .

The point C lies on OA and OA is perpendicular to BC.

The following diagram shows a circle with centre O and radius 40 cm.

The points A, B and C are on the circumference of the circle and $\mathrm{A}\stackrel{^<!--\; ^\; -->}{\mathrm{O}}\mathrm{C}=1.9\text{ radians}$.

A ship is sailing north from a point A towards point D. Point C is 175 km north of A. Point D is 60 km north of C. There is an island at E. The bearing of E from A is 055°. The bearing of E from C is 134°. This is shown in the following diagram.