Trigonometry (DP IB Analysis & Approaches (AA): SL): Exam Questions

Owen, Henry and Tom are rugby players passing a ball in a park. Owen is at point $\mathrm{O}$, Henry is at point $\mathrm{H}$ and Tom is at point $\mathrm{T}$, where $\mathrm{OH}=25$ m, $\mathrm{HT}=18$ m and $\mathrm{O}\hat{\mathrm{H}}\mathrm{T}=96°$.

(i) Draw and label a diagram to represent this information.

(ii) Find $\mathrm{OT}$.

Find $\mathrm{O}\hat{\mathrm{T}}\mathrm{H}$.

The players pass the ball within triangle $\mathrm{OHT}$.

Find the area of triangle $\mathrm{OHT}$.

A sailing race takes place on a large lake. The competitors must sail around five buoys, at the points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$, $\mathrm{D}$ and $\mathrm{E}$, in a clockwise direction.

$\mathrm{B}$ is due east of $\mathrm{A}$, $\mathrm{C}$ is due south of $\mathrm{B}$ and $\mathrm{E}$ is due south of $\mathrm{A}$. The bearing of $\mathrm{D}$ from $\mathrm{C}$ is 220°. $\mathrm{AB}=1200$ m, $\mathrm{BC}=600$ m, $\mathrm{CD}=800$ m and $\mathrm{EA}=1000$ m.

Draw and label a diagram to show the buoys $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$, $\mathrm{D}$ and $\mathrm{E}$, clearly marking the bearing and distances given above.

The boats start at $\mathrm{A}$. A support boat can travel directly across the course from $\mathrm{A}$ to $\mathrm{C}$ and from $\mathrm{A}$ to $\mathrm{D}$.

Find $\mathrm{AC}$.

Find $\mathrm{AD}$.

Find the bearing the support boat must follow to travel from $\mathrm{A}$ to $\mathrm{D}$.

The following diagram shows triangle $\mathrm{ABC}$, with $\mathrm{AC}=21$ km, $\mathrm{CB}=15$ km and $\mathrm{A}\hat{\mathrm{C}}\mathrm{B}=75°$.

Find the area of triangle $\mathrm{ABC}$.

Find AB.

Given that $\mathrm{C}\hat{\mathrm{A}}\mathrm{B}$ is acute, find $\mathrm{C}\hat{\mathrm{A}}\mathrm{B}$.

Triangle $\mathrm{ABC}$ has an area of 122 cm², where $\mathrm{AB}=24$ cm and $\mathrm{BC}=11$ cm.

Draw and label a diagram to show triangle $\mathrm{ABC}$, clearly marking the distances given.

Given that $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$ is acute, find

(i) $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$

(ii) $\mathrm{AC}$.

The quadrilateral $\mathrm{ABCD}$ shown below represents a farm paddock, where $\mathrm{AB}=246$ m, $\mathrm{BC}=312$ m, $\mathrm{AD}=257$ m, $\mathrm{D}\hat{\mathrm{A}}\mathrm{B}=96°$ and $\mathrm{B}\hat{\mathrm{C}}\mathrm{D}=78°$.

A fence is built from $\mathrm{B}$ to $\mathrm{D}$ to split the paddock into two parts.

Find the length of the fence, $\mathrm{BD}$.

Find the area of the paddock $\mathrm{ABCD}$.

A cliff 38 m high is perpendicular to the sea. The angle of depression from the top of the cliff to a boat at sea is 24°. A rock climber is partway up the cliff, and the angle of elevation from the boat to the climber is 14°.

Draw and label a diagram to show the top of the cliff, $\mathrm{T}$, the foot of the cliff, $\mathrm{F}$, the climber, $\mathrm{C}$, and the boat, $\mathrm{B}$, labelling all the angles and distances given above.

Find the distance from the boat to the foot of the cliff, $\mathrm{BF}$.

Find how far the climber must climb to reach the top of the cliff, $\mathrm{CT}$.

The following diagram shows triangle $\mathrm{XYZ}$, with $\mathrm{YZ}=5.4$ cm. The point $\mathrm{W}$ lies on $\left[\mathrm{XZ}\right]$, with $\mathrm{XW}=5.6$ cm, $\mathrm{WZ}=4.2$ cm and $\mathrm{YW}=5.8$ cm.

Find $\mathrm{Y}\hat{\mathrm{Z}}\mathrm{W}$.

Find the area of triangle XYZ.

Find the area of triangle XYW.

The distance between towns $\mathrm{X}$ and $\mathrm{Y}$ is 134.2 km, and the bearing of $\mathrm{X}$ from $\mathrm{Y}$ is 119°. Town $\mathrm{Z}$ is on a bearing of 207° from town $\mathrm{X}$, and is 54 km further south than town $\mathrm{X}$.

Draw and label a diagram to show towns $\mathrm{X}$, $\mathrm{Y}$ and $\mathrm{Z}$, clearly marking the bearings and distances given above.

Find $\mathrm{XZ}$.

Find $\mathrm{YZ}$.

$\mathrm{ABCD}$ is an isosceles trapezoid where $\mathrm{AB}=17$ m and $\mathrm{AD}=\mathrm{BC}=25$ m, as shown in the diagram below.

Find the height, $h$, of the trapezoid.

Find the area of the trapezoid.

The distance between Ho Chi Minh City and Hong Kong is known to be 1500 km. The bearing of Hong Kong from Ho Chi Minh City is 046°. Another city, Brisbane, is 6500 km from Ho Chi Minh City on a bearing of 136°. Calculate the distance between Hong Kong and Brisbane.

The diagram below shows an architect’s drawing of the front view of a house. The house is in the shape of a rectangle with a height of 10.8 m and has a roof in the shape of a right-angled isosceles triangle, $\mathrm{BCD}$, where $\mathrm{BD}=12.2$ m and $\mathrm{B}\hat{\mathrm{C}}\mathrm{D}=90$°. Next to the house is a garage in the shape of a rectangle measuring 4 m by 3.6 m, with a roof in the shape of a right-angled triangle, $\mathrm{EFG}$, where $\mathrm{GF}=4$ m and $\mathrm{E}\hat{\mathrm{F}}\mathrm{G}=37$°.

(i) Find $\mathrm{EG}$.

(ii) Find $\mathrm{BC}$.

Find the total area of the front view of the house.

A competitor is flying their kite in a competition. The kite is on a string of length 206 m and has an angle of elevation of 74° from the competitor, as shown in the diagram below.

Calculate the vertical height, in metres, that the kite is flying at above the point the competitor is holding it.

A second competitor raises their kite to the same vertical height from the same position as the first competitor. The angle between the two kites is 13°, as shown in the diagram below.

Calculate the length of the string for the kite flown by the second competitor.

A small airline operates between three locations A, B and C, in one particular country. B is located 530 km from A on a bearing of 248°. C is located 300 km due East from the midpoint, M, of [AB]. This information is shown in the diagram below.

Calculate AC.

Calculate the bearing that an aeroplane would need to fly on if it were travelling from C to B.

A gymnast is competing in the women’s uneven bars event. The bars are held in place by vertical supports at points A and B, as shown in the diagram, where A and B are situated at heights of 2.5 m and 1.7 m above the ground respectively. The horizontal distance between the bars is 1.1 m. This information is shown in the diagram below.

It can be assumed that the gymnast travels in a straight line when moving between points A and B.

Calculate the distance the gymnast travels in moving between points A and B.

Calculate the angle of depression from point A to point B.

When the gymnast is hanging vertically from the higher bar with her arms fully extended, there is a distance of 0.6 m between point A and her eye level.

Calculate the difference between the angle of depression calculated in part (b) and the angle of depression that the gymnast sees to point B.

The cross-section of a unicorn horn can be modelled by the triangle $\mathrm{ABC}$ shown in the diagram below, where $\mathrm{AB}=49$ cm and $\mathrm{BC}=58$ cm. The area of the cross-section is 168 cm².

Find $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$, the angle at the tip of the horn.

Find $\mathrm{AC}$, the length of the base of the horn.

The diagram below shows a quadrilateral $\mathrm{ABCD}$, where $\mathrm{B}\hat{\mathrm{A}}\mathrm{D}=59°$, $\mathrm{B}\hat{\mathrm{C}}\mathrm{D}=46°$, $\mathrm{AB}=14.4$ cm, $\mathrm{AD}=16.2$ cm and $\mathrm{BC}=19.7$ cm.

Find $\mathrm{BD}$.

Find $\mathrm{C}\hat{\mathrm{D}}\mathrm{B}$.

Show that the area of the quadrilateral is 235 cm², correct to the nearest cm².

Adah wants to estimate the height of a tree that stands at point $\mathrm{P}$ on the far bank of a river. The top of the tree is at point $\mathrm{Q}$, vertically above $\mathrm{P}$. The river is too dangerous for her to reach the base of the tree.

From point $\mathrm{M}$, the angle of elevation of the top of the tree is 20°. From point $\mathrm{N}$, on the edge of Adah's bank of the river, the angle of elevation of the top of the tree is 35°. The points $\mathrm{M}$, $\mathrm{N}$ and $\mathrm{P}$ lie on a horizontal straight line, and $\mathrm{MN}=12$ m.

This information is shown in the diagram below.

Find $\mathrm{NQ}$.

Find the height of the tree.

Adah borrows a boat and crosses the river in a straight line from $\mathrm{N}$ to $\mathrm{P}$. She travels at a constant rate of 50 metres every 15 minutes.

Find how long it takes her to cross the river.

The diagram below shows a triangular field $\mathrm{ABC}$ on a farm, where $\mathrm{AB}=17$ m, $\mathrm{AC}=45$ m and $\mathrm{B}\hat{\mathrm{A}}\mathrm{C}=38°$.

$\mathrm{X}$ is a point on $\mathrm{AC}$ such that $\mathrm{AX}:\mathrm{XC}=1:4$.

The field will be used for livestock, so a fence is to be put up around its perimeter.

Find the perimeter of the field.

The field is to be divided into two parts by a new fence from $\mathrm{B}$ to $\mathrm{X}$.

Find the area of triangle $\mathrm{BXC}$.

The diagram below shows a ship that is 86 m from an observation station, and a whale that has been spotted 137 m from the observation station.

The bearing of the ship from the observation station is 110°, and the bearing of the whale from the observation station is 072°.

Find the distance between the ship and the whale.

Find the bearing on which the ship must travel to reach the whale. Give your answer correct to 1 decimal place.

A trapezoidal prism, $\mathrm{ABCDEFGH}$, is shown in the diagram below. The length of the base is 7.5 cm and the width is 6.3 cm. The height of the prism is 6.5 cm and $\mathrm{BF}=8.8$ cm. In the trapezoidal cross-section $\mathrm{ABFE}$, $\left[\mathrm{AB}\right]$ is parallel to $\left[\mathrm{EF}\right]$.

Find $\mathrm{AB}$.

Find the size of $\mathrm{B}\hat{\mathrm{H}}\mathrm{A}$.

The diagram below shows the triangular sail $\mathrm{ABC}$ of a windsurfing board, with a horizontal boom $\mathrm{PC}$. $\mathrm{AB}=6.1$ m and makes an angle of 18° with the vertical. $\mathrm{BC}=4.7$ m and $\mathrm{B}\hat{\mathrm{C}}\mathrm{P}=70°$.

Find the area of the sail.

Find $\mathrm{PC}$, the length of the boom.

The area of triangle $\mathrm{ABC}$, shown below, is $12\sqrt{2}$.

Find the value of $x$.

Hence, find $\mathrm{BC}$.

Heron's formula gives the area of a triangle from its three side lengths, $a$, $b$ and $c$:

$\text{Area}=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$

where $s=\frac{a+b+c}{2}$ is half the perimeter of the triangle.

Verify that Heron's formula gives the area of triangle $\mathrm{ABC}$.

The diagram shows a triangular prism $\mathrm{ABCDEF}$ of height 18.2 cm, where $\mathrm{ED}=7.5$ cm, $\mathrm{EF}=5.3$ cm and $\mathrm{AC}=6.6$ cm.

$\mathrm{M}$ is the midpoint of $\mathrm{BC}$.

Find $\mathrm{DM}$.

Find $\mathrm{E}\hat{\mathrm{M}}\mathrm{D}$.

Find the area of triangle $\mathrm{EDM}$.

The diagram below shows a cable-stayed bridge crossing a river from $\mathrm{A}$ to $\mathrm{B}$. The embankment at $\mathrm{A}$ is 9.1 m above the horizontal river bed, and the embankment at $\mathrm{B}$ is 1.3 m above it. The river bed is 90 m wide.

A vertical column, $\mathrm{PV}$, of height 15 m stands at $\mathrm{P}$, the midpoint of the river bed. Two supporting cables run from the top of the column, $\mathrm{V}$, to $\mathrm{A}$ and to $\mathrm{B}$.

Find $\mathrm{V}\hat{\mathrm{B}}\mathrm{A}$, the angle between the supporting cable and the bridge span.

Find the total length of the two supporting cables.

A triangular piece of land has been marked out by placing string around 3 stakes at positions A, B and C, as shown in the diagram below. $\mathrm{AC}=22$ m, $\mathrm{BC}=14$ m and $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}=90$°.

Calculate the total length of the string used.

Calculate the area of the piece of land.

The section of land is to be adjusted. Points A and C remain fixed in position but point B is moved until $\mathrm{A}\hat{\mathrm{C}}\mathrm{B}=90$°. The overall length of the string does not change.

Calculate the new length of BC.

The shape ABCDEFG, as seen in the diagram below, shows the footprint of a new building that is to be constructed. ED and FG are parallel, as are CD, AG and EF. BC = 28 m, AB = 20 m, AG = 55 m, EF = 15 m and the perpendicular height of FG is 18 m. $\mathrm{B}\hat{\mathrm{A}}\mathrm{G}=90$°, $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}=65$° and $\mathrm{E}\hat{\mathrm{F}}\mathrm{G}=58$°.

Calculate the area of the footprint of the building.

An internal wall is to be constructed along [DG].

Find the length of the internal wall and the angle it makes with [FG].

A security lamp is situated at a height of 2.5 m and positioned so that the central axis of the light bulb is directed perpendicularly to the horizontal. When the lamp is switched on the light spreads out in all directions up to an angle of 38° from the central axis of the light bulb. This information is shown in the diagram below.

Calculate the horizontal distance on the floor that is illuminated by the lamp.

The area illuminated is not sufficient so the lamp is repositioned at the same height so that the central axis of the light bulb is now at an angle of 70° from the horizontal.

Calculate the percentage increase in the horizontal distance that is now illuminated.

An equilateral triangular jigsaw piece has an edge length of 32 mm. Several of these pieces are connected together with the vertices of the triangular pieces alternately pointing up and then down. The completed jigsaw puzzle is in the shape of a parallelogram with a side length of 64 cm and a perpendicular height of $24\sqrt{3}$ cm. A diagram illustrating this information can be seen below.

Calculate the number of individual jigsaw pieces in the puzzle.

A second jigsaw is to be designed using 289 of the same type of individual pieces. The completed puzzle will this time be in the shape of an equilateral triangle.

Find the number of pieces along each side of the triangle.

A roof with a symmetrical triangular cross-section, ABC, is being designed for the top of a building. The horizontal width that the roof must span is 28 m and the lengths of the timbers used for the angled part of the cross-section are 21 m, as shown in the diagram below.

Calculate $\mathrm{C}\hat{\mathrm{A}}\mathrm{B}$.

An alternative design idea for the roof is to shorten AC and to make the apex of the roof a right angle. BC remains the same length as it was originally. These changes can be seen in the diagram below. The point X is situated such that it is directly beneath point C.

Calculate the new length of AC.

Calculate the vertical height CX of this alternative design for the roof.

A bird is perched on the edge of a building with its eye at a height of 9.5 m above ground level. A person is holding a sandwich at a height of 1.2 m from the ground and the distance between the ground and the person’s eye level is 1.6 m. A diagram showing this is below.

The bird sees the sandwich at an angle of depression of 52°.

Calculate the distance that the bird must fly to reach the food.

The person’s eyes are 0.3 m further away from the building than the sandwich.

Find the angle of elevation at which the person sees the bird.

A second bird is perched on a lamp post on the other side of the person at a horizontal distance of 5 m. The person sees this bird at an angle of elevation of 37°.

Find the vertical distance between the two birds.

A wheelchair ramp is required to provide access to a building with a door that is located 22 cm above ground level. The maximum angle that a ramp must be from the horizontal is 4.8°.

Calculate the minimum horizontal distance that the ramp must extend out.

The wheelchair ramp is built using the minimum distance found in part (a), rounded to 3 significant figures. The ramp is supported by a steel frame, a cross section of which can be seen in the diagram below. A metal strut joins M, the midpoint of [AC], to a point X on [AB]. XM = 11.1 cm and $\mathrm{M}\hat{\mathrm{X}}\mathrm{C}=90$°.

Calculate XB.

The following diagram shows four islands, $\mathrm{P}$, $\mathrm{Q}$, $\mathrm{R}$ and $\mathrm{S}$, where $\mathrm{PQ}=8.5$ km, $\mathrm{QR}=16.2$ km, $\mathrm{RS}=12.5$ km, $\mathrm{P}\hat{\mathrm{Q}}\mathrm{S}=25°$ and $\mathrm{Q}\hat{\mathrm{R}}\mathrm{S}=82.1°$. Island $\mathrm{Q}$ is due north of island $\mathrm{P}$.

Mark makes deliveries around the islands. He travels from $\mathrm{Q}$ to $\mathrm{S}$, then from $\mathrm{S}$ to $\mathrm{P}$, and finally from $\mathrm{P}$ to $\mathrm{R}$.

Find the total distance Mark travels.

The diagram below shows a pyramid $\mathrm{ABCDE}$ with a rectangular base $\mathrm{ABCD}$, where $\mathrm{DC}=5.9$ cm, $\mathrm{AD}=3.7$ cm and $\mathrm{AE}=7.4$ cm. The vertex $\mathrm{E}$ is directly above the centre of the base.

Find $\mathrm{A}\hat{\mathrm{E}}\mathrm{C}$.

$\mathrm{P}$ is a point on the edge $\mathrm{EB}$ such that $\mathrm{EP}:\mathrm{PB}=1:4$.

Find the area of triangle $\mathrm{EPD}$.

The diagram below shows a police helicopter at point $\mathrm{B}$ using a beam of light to search an area of horizontal ground between $\mathrm{A}$ and $\mathrm{C}$. The edge of the beam furthest from the helicopter, $\mathrm{BC}$, is 22 m long, and the angle of depression from the helicopter to $\mathrm{C}$ is 47°.

The area of the cross-section of the beam, triangle $\mathrm{ABC}$, is 23 m².

Find $\mathrm{AC}$, the length of ground lit by the beam.

Find $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$, the angle of the beam.

A piece of playground equipment has two ropes fixed to a hook at point $\mathrm{A}$ on the edge of a gap. The ropes are pulled taut across the gap and fixed at points $\mathrm{B}$ and $\mathrm{C}$ on the other side. The left embankment is 2.3 m high. $\mathrm{B}$ is at the top of the right embankment, and $\mathrm{C}$ is vertically below $\mathrm{B}$, 0.8 m above the ground. The angle between the ropes is 23°, and the horizontal width of the gap is 1.4 m. This information is shown in the diagram below.

Find $\mathrm{BC}$.

A third rope, of length 0.9 m, is fixed at $\mathrm{B}$ and at a point $\mathrm{P}$ on the rope $\mathrm{AC}$.

Find $\mathrm{B}\hat{\mathrm{P}}\mathrm{C}$ and hence find $\mathrm{PC}$.

A pitched roof is made from a timber frame, $\mathrm{ABCDEF}$. Its horizontal rectangular base $\mathrm{ADFC}$ measures 15.3 m by 8.2 m. The ridge $\mathrm{BE}$ is parallel to $\mathrm{AD}$, and its midpoint is directly above the point where $\left[\mathrm{AF}\right]$ and $\left[\mathrm{CD}\right]$ intersect. $\mathrm{BE}$ is 12.1 m long and is 2.2 m above the base $\mathrm{ADFC}$.

Find the total length of timber needed for the frame.

An internal beam runs from $\mathrm{M}$, the midpoint of $\mathrm{AC}$, to $\mathrm{E}$.

Find $\mathrm{ME}$.

In the diagram below, $\mathrm{AB}$, $\mathrm{BC}$ and $\mathrm{AC}$ are steel beams on the first floor of a building under construction. Triangle $\mathrm{ABC}$ lies in a horizontal plane 5 m above the ground, with $\mathrm{AB}=7.9$ m, $\mathrm{BC}=5.8$ m and $\mathrm{AC}=8.3$ m. Viewed from above, the beam $\mathrm{AB}$ is on a bearing of 078° from $\mathrm{A}$.

A pot of paint has been left on beam $\mathrm{BC}$ at $\mathrm{M}$, halfway between $\mathrm{B}$ and $\mathrm{C}$.

Find the bearing of the pot of paint from $\mathrm{A}$.

A workman stands on the second floor of the building, directly above $\mathrm{A}$, with his eyes 12 m above the ground.

Find

(i) the angle of depression,

(ii) the distance

from the workman's eyes to the pot of paint.

A tent has a symmetrical triangular cross-section, $\mathrm{ABC}$, and stands on horizontal ground. The perpendicular height of the tent is 1.2 m. Guy ropes are attached at points $\mathrm{X}$ and $\mathrm{Y}$ on $\mathrm{AB}$ and $\mathrm{BC}$, where $\mathrm{BX}=\mathrm{BY}=0.7$ m. Each guy rope is 1.1 m long, and they are fixed to the ground at points $\mathrm{P}$ and $\mathrm{Q}$ so that $\mathrm{P}\hat{\mathrm{X}}\mathrm{A}=\mathrm{Q}\hat{\mathrm{Y}}\mathrm{C}=30°$. The points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$, $\mathrm{P}$, $\mathrm{Q}$, $\mathrm{X}$ and $\mathrm{Y}$ all lie in the same vertical plane, as shown in the diagram below.

Find $\mathrm{PQ}$, the distance between the points where the guy ropes are fixed.

The diagram below shows a door wedge, $\mathrm{ABCDEFGH}$. $\mathrm{ADEH}$ is a horizontal surface and the angles $\mathrm{G}\hat{\mathrm{H}}\mathrm{D}$ and $\mathrm{F}\hat{\mathrm{E}}\mathrm{A}$ are right angles. The face $\mathrm{ABCD}$ is a square face parallel to $\mathrm{EFGH}$, with the midpoints of $\left[\mathrm{AD}\right]$ and $\left[\mathrm{EH}\right]$ aligned. $\mathrm{FG}=12$ cm, $\mathrm{GH}=7$ cm, $\mathrm{DH}=15$ cm and $\mathrm{AD}=2$ cm. This information is represented in the diagram below.

Find the size of the angle $\mathrm{C}\hat{\mathrm{G}}\mathrm{H}$.

Calculate the length AG.

(i) Find the perpendicular distance between $\left[\mathrm{BC}\right]$ and $\left[\mathrm{FG}\right]$.

(ii) Hence find the angle that the plane $\mathrm{BCFG}$ makes with the horizontal.

A spider spins a web in a single plane. The web has 7 vertices, $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$, $\mathrm{D}$, $\mathrm{E}$, $\mathrm{F}$ and $\mathrm{G}$, equally spaced around a centre point $\mathrm{O}$. Threads of equal length join each pair of neighbouring vertices ($\mathrm{AB}$, $\mathrm{BC}$, and so on), and a thread joins each vertex to the centre ($\mathrm{OA}$, $\mathrm{OB}$, and so on). This is shown in the diagram below.

The spider then adds a thread from each vertex to the midpoint of the next thread to the centre, moving clockwise around the web; for example, from $\mathrm{G}$ to the midpoint of $\mathrm{OA}$.

The spider can produce 220 cm of silk in a day.

Find the maximum possible length of $\mathrm{OA}$ if the spider is to complete the web in one day.

The diagram below shows a funnel in the shape of a right cone with a smaller cone removed from its end. The circular faces at the two ends are parallel. The perpendicular height of the complete cone is 168 mm. The diameter of the funnel is 98 mm at its upper end and 7 mm at its lower end.

A grain of sugar is left at point $\mathrm{P}$, $\frac{1}{3}$ of the way up the slanted height of the funnel. An ant sits at point $\mathrm{A}$ on the top edge of the funnel. $\left[\mathrm{AB}\right]$ is a diameter of the larger circular face, and the points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{P}$ and the axis of the cone lie in a single plane.

Find $\mathrm{AP}$, the direct distance between the ant and the grain of sugar.

Find the angle of depression from the ant to the grain of sugar.

A pendant for a necklace is made in the shape of a symmetrical hexagon, $\mathrm{ABCDEF}$, as shown in the diagram below. $\mathrm{AF}$, $\mathrm{BE}$ and $\mathrm{CD}$ are parallel, $\mathrm{BE}=22$ mm and $\mathrm{AF}=\mathrm{CD}=16$ mm. $\mathrm{AB}=\mathrm{BC}$ and $\mathrm{EF}=\mathrm{DE}$. The total width of the pendant is 54 mm.

The shaded triangles $\mathrm{AFE}$ and $\mathrm{CDE}$ are made of silver, and the rest of the pendant is made of gold.

Find the percentage of the area of the pendant that is silver.

The 'H' on the Hollywood sign is 13.7 m high, measured along its rear face. Each leg is 3 m wide and the gap between the legs is 4 m. The cross bar is as wide (measured from top to bottom in the diagram) as each leg, and it is centred on the height of the 'H'. This information is shown in the diagram below.

The 'H' stands on horizontal ground but is tilted backwards, so that its rear face makes an angle of 5° with the vertical. During repairs, a metal support bar is fixed between a point $\mathrm{A}$ on the ground and the point $\mathrm{M}$, the midpoint of the rear of the cross bar. $\mathrm{B}$ is the midpoint of the gap between the legs at ground level. The plane containing $\mathrm{A}$, $\mathrm{M}$ and $\mathrm{B}$ is perpendicular to the rear face of the 'H', and the angle between the support bar and the rear face, $\mathrm{A}\hat{\mathrm{M}}\mathrm{B}$, is 30°.

Find the length of the support bar.

More support bars are needed, from the ground to the midpoint of the top of the rear face of each leg. These supports meet the ground at the same point, $\mathrm{A}$, as the first support bar.

Find

(i) the length of one of these supports,

(ii) the angle that it makes with the horizontal.