Mary (M) sits on the top of the Eastern side of9,r*sz;sbod,zu c6 jl the Grand Canyon. Below her stands Peter (P), at the bottom of the canyon. It is known that the depth of the Grand Canyon is 1850 meters at thilz; db,u6czj9s rs, *os point. Nathan (N) stands on the Western side of the bottom of the Canyon, with a distance of 2.5 km to Peter.


1.Write down the depth of the Grand Canyon, MP, in kilometers.MP=  km
2.On the diagram, label the angle of elevation from N to M with an x.
3.Find the size of the angle of elevation from N to M.  $^{\circ}$
4.Find MN, the distance in kilometers between Mary and Nathan.  km

  A ladder 3.7 m long leans against a vertical wall. The distance from v xg,fw1 1r k)46oy9+(jccqb6ccsf3 iy v7v eothe top of the ladder tobxf,y fq 7 vo1vi 4c6jr3 1 o9+)vek(gcccswy6 the ground is 3.1 m.


1.Represent this information on a diagram in the space provided above. Show the ground, the ladder and the wall as the labelled line segments.
2.Find the distance from the bottom of the ladder to the wall.  m
3.Write down the acute angle made by the ladder with the wall.  $^{\circ}$

  The diagram below shows aphp219 jmeo( q triangle ABC. The size of angle $\mathrm{A}\hat{\mathrm{M}}\mathrm{C}$ is $150^{\circ}$, where M is the midpoint of AB. MBC is an isosceles triangle, with MC = MB = 4 cm.

1.Write down the length of AM.  cm
2.Find the size of angle $\textrm{C}\hat{\textrm{M}}\textrm{B}$.  $^{\circ}$
3.Find the size of angle $\textrm{A}\hat{\textrm{B}}\textrm{C}$.  $^{\circ}$
4.Find the length of CB.  cm

  A lawn sprinkler wate9kwy zz9)g a*zu4w8gfring a garden sprays at an arc of $140^{\circ}$ with a maximum spray distance of 3.5 metres, as shown in the diagram below.

1.Calculate the area of lawn that is watered by the sprinkler.  $m^2$
2.Calculate the length of the outer arc, where the maximum spray
distance lands.  m

  In a school playground, the grass is watered by lawn j zv- ak/1nwd5sprinklers that are set to spray -j1kw/ v5dzan at an arc of $220^{\circ}$ with a maximum spray distance of 55 metres, as shown in the diagram below.


1.Calculate the area of grass that is watered by this sprinkler.  $m^2$
2.Calculate the length of the sprinkler arc edge, where the maximum
spray distance lands.  m

  A windscreen wiper bladetcf21,ctxu 1 a/1dx ms has a length of 0.45 m. When it's raining and the blade is in motion, it moves through an afs1x1x c ma d,12c/tutrc angle of $\dfrac {3\pi} 4$ radians.


1.Determine the area of the windscreen wiped by the blade.  $m^2$
2.Determine the arc length travelled by the outer edge of the blade.  m

  There are two traffic lights on a road below a 120 m hi h zmr 4-g,xsawm0nl49 v9zja.gh skyscraper. From the top of the skyscraper, the traffic lights can h,4lg94 szv.0-nm x a9mwzarjbe seen at angles of depression of $35^{\circ}$ and $65^{\circ}$ on each side of the skyscraper, as illustrated in the diagram below.


Find the distance between the two traffic lights, correct to the nearest whole metre.  m

  A triangular plot of farming land is sdltec7zf; 2 o (r2yc4fhown in the diagram below. The longest side is 120120 m, witd c(ozfel2t4yr2; fc7 h the two adjoining angles to this length being $35^{\circ}$ and $25^{\circ}$.


1.Find the lengths of two other sides of this plot of land.a$\approx$  m,b$\approx$  m
2.Find the area of the plot of farming land.  $m^2$

  A national park is in the form of a triangle, ABC, as shown in the following diaw:8hi4ns af7 ngram. Side length ABnwns4f:i8 ah 7 is 49 km and side length BC is 66 km. Angle $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$ is $58^{\circ}$.


1.Calculate the side length AC in km.  km
2.Calculate the area of the park.  $km^2$

  John is building furniture using cylindrical logs of length 1rs+g1 4 odgmx+.8 m and radius 9.2 cm. A wedge is cut from one log and the cross-section of this log is r+1xos4gg+ md illustrated in the following diagram.


1.Find the length of the wedge arc, ABC.  cm
2.Find the area of the empty sector, OABC.  $cm^2$
3.Find the volume of each log.  $cm^3$

  A farmer has a field in the form of a triangl il.;p7-tzm0 rh-fyqie, ABC, as shown in the following diagram. Side length AB is 150 m and side length BC is 200 m l0y fi-- .mpitrz;7hq. Angle $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$ is $62^{\circ}$.


1.Calculate the side length AC.  m
2.Calculate the area of the field, rounding your answer to the nearest square meter.  $m^2$

  During a hiking trip, Jim stopped and checked his comu nvsele 0, ihd(7iakpd)1y- 1pass for directions. The compass has a circular face with centre at O and the diameter of 20 cm. The needle of the compass is at an ayi v017iued s(h,lp-e) k1ndangle of $\theta$ to the East of North, covering an arc length of $l$, as illustrated in the diagram below.

1.Determine the circumference of the face of the compass.  cm
2.Using the diagram, write down the size of the angle $\theta$ in degrees.  $^{\circ}$
3.Calculate the arc length, $l$.  cm
4.Calculate the total area of the shaded sectors on the diagram.  $cm^2$

  A rotating sprinkler waters part of a lawn, shown as the shad3qtt2h j)6exjed area in the diagram below. The sprinkler covers a distance of 13m, reaching a maximum distance of 17m and cov)t qjhj 6e3xt2ering an angle of $130^{circ}$.

1.Calculate the length of the outer arc the sprinkler can reach, the arc from point A to point B.  m
2.Find the area the sprinkler can cover. Give your answer to the nearest square metre.  $m^2$

  A surveillance camera records the area in front of a warehousevj 0cx3)hxu4zet 6c:nncs8a., shown as the shaded area in the f6su az:c0 )xc.38vntn c xh4ejollowing diagram. The camera's field of view has a radial distance of 15m, reaching a maximum distance of 21m away from the camera, and covering an angle of $150^{\circ}$.




1.Calculate the length of the outer arc the camera can record, the arc from point A to point B.  m
2.Find the area the camera can record.  $m^2$

  Cylindrical logs of length 2.7 m and radius 7.5 c7w:h7th9kcrz z2w ic87j: dvc -d9fehm are used to build a wooden cabin. A wedge is cut from one log and the cross-section of this log is illustrated in the follhfhtz9d7 ivrcz cdj9 -ww:e :8hk7c27 owing diagram.


1.Convert 108 degrees into radians, leaving your answer in exact values.
2.Find the length of the missing arc ABC.  cm
3.Find the area of the missing sector ABCO.  $cm^2$
4.Find the volume of this log.  $cm^3$

  A right pyramid has apex V and solz 2425osa8iw zjmy/quare base ABCD. The vertical height of the pyramid, VM, is 5 cm. The slopijm o2 swz/ 4yz8a5l2iong edges are 8 cm long.


1.Calculate the length of MC.  cm
2.Calculate the size of the angle that sloping edge VC makes with the
vertical height of the pyramid.  $^{\circ}$
3.Calculate the area of the triangle ABC.  $cm^2$

  A right pyramid has apex V and square base ABCD with the base diagonal ACko4g7 we+8z y:l6youe =4cm. The sloping edges ayzey8:w o6+ eglko74 ure 7 cm long.



1.Calculate the size of the angle that sloping edge VC makes with
the pyramid base.  ${\circ}$
2.Calculate the vertical height of the pyramid, VO.  cm
3.Calculate the volume of the pyramid.  $cm^3$

  Triangle Island is a small island located off the north-western rk7* qs6f-jnwtip of British Columbia, Canada. Two6k rj7-q*wsfn side lengths of Triangle Island are equal to 1600 and 1800 metres and the angle between these two sides is equal to $78^{\circ}$.

1.Calculate the length of the third side of Triangle island.  m
2.Find the area of a Triangle Island in $m^2$. Give your answer in the form $a\times 10^k$, where $1 \leq a < 10$ and $k \in \mathbb Z$.  $\times10^6m^2$

  The quadrilateral ABCD has side lengths AB =17 cm, BC = 12cm, and CD = 11c.sc*2mw:v q so5s0: .rjgt gul lo;y;jm. The size ofc.tus s*0.jwo5;;goglvqlj2mr: s:y angle $\mathrm{A}\hat{\mathrm{D}}\mathrm{C}$ is $105{\circ}$ and the size of angle $\mathrm{C}\hat{\mathrm{A}}\mathrm{D}$ is $25^{\circ}$.


1.Find the length of AC in centimetres.  cm
2.Find the size of angle $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$.  $^{\circ}$

  The diagram below shows quadrilateral ABCD with side lengths of AD = 12 da6 . -dg9oyq7tlr(efm, DC = 7 m and BC = 5 mrf(at-dod9qyel.7 6 g . Angle $\mathrm{B}\hat{\mathrm{A}}\mathrm{D}=50^{\circ}$ and angle $\mathrm{B}\hat{\mathrm{C}}\mathrm{D}=100^{\circ}$.


A new line is drawn from B to D direct.
1.Calculate the length of this new line, BD.  m
2.Calculate the size of angle $\mathrm{A}\hat{\mathrm{B}}\mathrm{D}$.  $^{\circ}$

  Cylindrical logs of length 2.2 m and radius 7.6 cm are used to construct a f jw:u s zsl:k1yb(86ktence line. Part of the logs are remolky w6:us s:j1(t 8kzbved, as illustrated in the diagram below.

Find the volume of each log.  $cm^3$

  Consider the analogue clock below. The clock has a ht+2agcp-lh6diameter of 30 cm and is shaded between 1 and 6. ca2+ gphl6-h t


1.Find the length of the arc between 1 and 6, rounded to the nearest cm.  cm
2.Find the area of the sector between 1 and 6, rounded to the nearest
integer.  $cm^2$

  A bakery has designed a box to deliver their spx j:z :qna8vm rp jl+-mw0-o7feciality birthday cakes. The box has a rectangular prism base, with a rectangular pyramid top, as illustrated in the diagram below with dimensiow0+7x jj:l voamf:qz8--n pmrns shown.
After the cake is put into the box, the outside of the box is completely wrapped in colourful gift paper.

Point O shown on the diagram is the midpoint of the base of the pyramid.
1.Find the value of a shown in the diagram; the slant height of one of the pyramid faces.  cm
2.Find the total outer surface area of the box to be wrapped in gift paper.  $cm^2$

  Joe creates a small baseball field in his backyard to play with *zy0la aetnd gw)0/q5 his children. The field is covered by grass in the shape of a sector of a circle with radius 8 metres, as shown in the following diagram. The angle at the cealgtna)/z *0 q5ey w0dntre of the sector is $\theta^{\circ}$


A total of $48 m^2$ of grass was used to cover the field.
1.Find the value of $\theta$.  $^{\circ}$
Joe asks his son, Lancer, to run around the field twice to warm up before playing a game.
1.Find the total distance that Lancer runs.d=  m

  Kindergarten students are cv qt()*1niuij gt .h49gb8bx6dl s2koreating cone-shaped birthday hats from pieces of colorful paper circles. The first step is to cut out a sector of the paper, as illustrated by sector OABC in txb.vjs29*8b6i1i odg ) 4utl qngt( hkhe diagram below. The paper circles have centre O and . The length of arc ABC is $4\pi$ cm and angle $\mathrm{A}\hat{\mathrm{O}}\mathrm{C}= \dfrac {2\pi} 7$

1.Find the radius of the paper circles, r.  cm
2.Find the area of the sector the students cut out, OABC.  $cm^2$

  A right pyramid has apex V and square base ABCD, withe ,1 t9enhs/vr AB = 3 cm. The vertical heighht9s re 1n,ve/t of the pyramid, VM, is 12 cm.


1.Calculate the length of VA.  cm
2.Calculate the volume of the pyramid.  $cm^3$

  The sides of a triangle ABC have the following lengths AB =a-wv4tt . d1ce6 jm)wt 40 mm, BC = 30 mm, and)w1t tvwce6mdj a -t4. CA = 55 mm.



1.Calculate the size of the angle $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$.  $^{\circ}$
2.Calculate the area of the triangle ABC.  $mm^2$
A point D is located outside of the triangle, such that angle $\mathrm{D}\hat{\mathrm{A}}\mathrm{B}$ is $80^{\circ}$ and angle $\mathrm{D}\hat{\mathrm{B}}\mathrm{A}$ is $10^{\circ}$.
3.Show that the angle $\mathrm{A}\hat{\mathrm{D}}\mathrm{B}$ is $90^{\circ}$.  $^{\circ}$
4.Find the distances AD and BD.  mm
5.Find the shortest path which starts at point B and passes through all points A, C, D (in any order).  mm

  Two hot air balloons, located at points A and B, float in t4x9gu,qcde i7 kjw4cnfc 30j +he sky 50 m apart from each other on the same height above g4cj7ex3+w0n4i,c9d uk fc jqa flat (horizontal) ground. Two straight ropes, AP and BP, are attached to the same point, P, on the ground. The length of AP is
36m and angle $\mathrm{A}\hat{\mathrm{B}}\mathrm{P}$ is $45^{\circ}$.


1.On the diagram, draw and label with an $\alpha$ the angle of elevation of A Angle APB is acute.
2.Find the size of angle $\mathrm{A}\hat{\mathrm{P}}\mathrm{B}$.  $^{\circ}$
3.Find the size of the angle of elevation of A from P.  $^{\circ}$

  A plot of farmland is shown by the shaded region in the diagram below. The plo v+tkxa-awh.ylv 5. 1m5akxu,t has a curved border which can be described as an arc, PQ, of a cir k +- 1aytu.ax5almw,hx5v.vkcle with centre O and radius,$r=500m$, such that $PÔQ =80^{\circ}$ The straight border of the farmland is defined by the chord PQ.


The farm owner wishes to construct a fence along the curved border of the plot.
1.Find the length of the fence to be built.  m
2.Find the area of the farmland, rounding your answer to the nearest $m^2$.  $m^2$

  A large triangular garden, ABC, is m3400muanzgn shown in the diagram below. A walking path passes through the garden at point D and leaves at point E, such that DE is para3ma0 nn zm0ug4llel to the garden boundary BC.


Triangles ABC and ADE are found to be similar (same shape, but side lengths can vary). Angles $AD̂E = AB̂C =\theta$ and $AÊD = AĈB =\alpha$.
1.Find the value of
1.x.  m
2.y.  m
2.Find the value of
1.$\alpha$.  $^{\circ}$
2.$\theta$  $^{\circ}$
3.$\gamma$  $^{\circ}$
3.Find the area of the triangle ABC, rounding your answer to the nearest $m^2$.  $m^2$

  Two concentric circles with centre O are sho hrxe6bg4y2 n7.*tq5tz+3kkn z9ng z twn in the diagram below. The radius of the smaller circle is 5 cm and the radius of the larger circle is 7 cm. Poinn.ztgz zgtt2 bk*n3k74 exq n9r56hy+ts A and B lie on the circumference of the larger circle such that angle $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}$ is $\frac {\pi}{5}$ radians.


1.Find the length of the short arc AB.  cm
2.Find the area of the shaded region.  $cm^2$

  In a triangle ABC,AB=3 cm,BC=5 cm and 33ktp juvi j5rya+1h 9$\mathrm{A}\hat{\mathrm{C}}\mathrm{B} = \dfrac{\pi}{6}A$.
1.Find, to three significant figures, the two possible lengths of [AC].  cm   cm
2.Find the difference between the areas of the two possible triangles ABC.  $cm^2$

  In a triangle ABC,AB=12 cm and AC=15 cm. The area of thlzsxztbqe j58ekt51zw.+( .s 0w:ph ye triangle is $80cm^2$.
1.Find the two possible values for angle BÂC.  $^{\circ}$,  $^{\circ}$
2.Given that BÂC is obtuse, find the length of side BC.  cm

  A triangle ABC has side lenqege 1 9s xu14h+k+ch.rnm p5wgths a = 10.2 cm, b= 17.5 cm and an area of $32 cm^2$. Find the largest possible perimeter of triangle ABC.$P{_\Delta ABC}$=  CM

  A toy house has a roof in the shape of a right-angled isosceles triangle, -ygtbd9l1c27( d9u y-; sflahqhocz*g:mfl. ABC.AB = 6 cm and angle do gqmhd.lg:-ta 7fy9 l h2*l(y1;bzf-c c9s u$\textrm{A}\hat{\textrm{C}}\textrm{B}=90^{\circ}$.


1.Find the length of AC.  cm
2.Calculate the area of triangle ABC.  $cm^2$
The garage of the toy house also has a roof in the shape of right-angled triangle, LMN. , angle $\textrm{L}\hat{\textrm{M}}\textrm{N}=56^{\circ}$ and angle $\textrm{N}\hat{\textrm{L}}\textrm{M}=90^{\circ}$.
3.Calculate the length of MN.  cm

  From a sailboat moored at sea, the anglzgn+u-(fk5l r+c x +we 0rz-nbvtz .;le of elevation to the top of a 20 m high c.nl zr-b (gvr -x 5fweu+l; ++tzck0znliff is $22^{\circ}$. On the cliff top there is a coconut tree. The angle of elevation to the coconut tree is $29^{\circ}$.
1.Draw a diagram representing the situation.
2.Find the height of the coconut tree.  m

  The quadrilateral ABCD shown below represents a sports compluu y:/1s5b/svcahfy lzr v1 9:ex, where AB = 210 m, BC = 90 m and CD = 130 mrlu :19aby/vs/f15vhczs: uy . Angle $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$ is $70^{\circ}$ and angle $\mathrm{C}\hat{\mathrm{D}}\mathrm{A} is $105^{\circ}$.


A straight path through the sports complex joins points A and C.
1.Find the length of the path AC.  m
2.Show that the angle $\mathrm{D}\hat{\mathrm{A}}\mathrm{C}$ is $39.4^{\circ}$ , correct to three significant figures.  $^{\circ}$
3.Find the area of the sports complex.  $m^2$

A new path, DE, is to be built such that E is the point on AC closest to D.

4.Find the length of the path DE.  m

The section of the sports complex represented by triangle CDE will be an outdoor running track. A track will be marked along the sides of this section.

5.Calculate the total length of the track.  m

  The quadrilateral ABCD is shown be h+ ; :)w4wye 3rv4rljm5juovqlow, where AB = 56m, AD =122 m, CD =40 m, and the angleshr5yj qw +u;): e3jvl 44wvomr $\mathrm{B}\hat{\mathrm{A}}\mathrm{D}$ and $\mathrm{A}\hat{\mathrm{D}}\mathrm{C}$ are $68^{\circ}$ and $80^{\circ}$ respectively.


1.Find the length of line AC.  m
2.Find the area of triangle ACD.  $m^2$
3.Find the angle $\mathrm{D}\hat{\mathrm{A}}\mathrm{C}$. Give your answer in degrees correct to one
decimal place.  $^{\circ}$
4.Find the shortest distance between point B and the line AC.  m
5.Find the length of the perimeter of triangle ABC.  m

  A flagpole stands at the top of a building. The anxu lisl:5e- l3gle of elevation from a point on thelsi5 u3 lx:-el ground 40m from the base of the building to the bottom of the flagpole is $29^{\circ}$, and to the top of the flagpole is $31^{\circ}$.
1.Draw a diagram representing the situation.
2.Find the length of the flagpole.  m

  Consider the triangle below, which hrd9(t smt4* s l9 vynqia*ttgv c7 a0787jm0qpas side lengths AB = 25 cm and BC = 35 cm. The angle adjacsc0vmygiq89qt7 a*dtajmt ts n9l(vrp 0 477*ent to these side lengths, AB̂C, is $70^{\circ}$.

1.Find the area of the triangle, rounding your answer to the nearest $cm^2$.  $cm^2$
When drawing the triangle, George measured each side length correct the nearest centimetre, and angle AB̂C correct to the nearest degree.
2.Calculate the maximum possible area of the triangle, rounding your answer to the nearest $cm^2$.  $cm^2$

  The angle of depression from a lookoub9e;iir wb/o nwob kt,6( yvf-84y ym-t at the top of a lighthouse (A) down to a boat located at po wb-ike/w(f6yrtmb4;noiy by8- 9ov,int C is $30^{\circ}$.
The boat travels towards the lighthouse and after 1 minute has travelled a distance of 50 m and is now located at point B. The angle of elevation from the boat at B up to the lighthouse lookout is $60^{\circ}$.


1.Write down the value of $x^{\circ}$ in the diagram.  $^{\circ}$
2.Find the height of the lighthouse.  m
3.Find the speed of the boat, in metres per second, from C to B.  $ms^{-1}$

  Melody is camping with her friends in a forest. After s)j ao(gks,93nyx z* noetting up the tent, she goes for a walk t*,(nno9gxs 3 ykaoz)j o see the nearby surroundings. She walks 200 m on a bearing of $280^{\circ}$, changes direction, then walks another 150 m on a bearing of $035^{\circ}$.
1.Find the distance Melody is from the tent at this point.  m
2.Determine the bearing Melody should walk on to return to the tent.  $^{\circ}$

  On a trip to Morocco, a group of tourists experience a horse trek through the Sa.9j:kxwcrw8u v9we d(haran desert. They start from a horse f dervk:9uw (.j wcx9w8arm, moving on a bearing of $075^{\circ}$ for 3.2km to reach an oasis. After taking a rest there, they take another ride on a bearing of $210^{\circ}$ for 2.1km to visit the Draa valley.
1.Find the distance that they need to cover to get back to the farm.  km
2.Determine the bearing they should ride on to get back to the farm.  $^{\circ}$

  Consider a triangle with sidesAB=10 3u0 sod4yyp8dmm and AC=8 mm. Given that the area of the triangle u8doy3p4dy0 sis $24 mm^2$, find the possible values for the length of side BC.BC$\approx$  mm and  mm

  Daniela (D) stands near the window on the x-,sn4na 3:bxzk2xy2uetva *second floor of the six-story apartment building and can see Andrea (A), who is 14 m from the base of the building. The angle of depression from Dan-zu n x xe ,2*nvx4:2ab3katysiela to Andrea is $20^{\circ}$.


1.Calculate the distance from Daniela to Andrea, AD.  m
Clara (C) is standing at the window of the sixth floor, three times higher from the ground than Daniela.
2.Calculate the angle of depression from C to A.  $^{\circ}$

  A triangular farm is shown in the diagram b g6y dh 1y;3o5dsagi9yelow. A river runs along the edge from A to B and the farm house is located at C, a safe distance from the river. The distance from the farm house to point A is 270 m and to point B is 360 m. The angle 6 hdys5;9goy3aiyd g1$\mathrm{A}\hat{\mathrm{C}}\mathrm{B}$ is $130^{\circ}$.



1.Calculate the distance along the river from A to B.  m

The cost of fencing is 130130 Australian dollars (AUD) per metre.

2.alculate the total cost of fencing the whole perimeter of the farm, providing your answer to the nearest dollar.  
3.Find the sizes of angels $\mathrm{C}\hat{\mathrm{A}}\mathrm{B}$ and $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$.  $^{\circ}$
4.Calculate the area of the farm. Give your answer correct to the nearest whole square metre.  $m^2$
5.Calculate the shortest distance from the farm house to the river.  m

A tower is located at point B. The angle of elevation to the top of the tower, H, from point C is measured to be $1^{\circ}$.

6.Calculate the vertical height, BH, of the tower, correct to the nearest centimetre.  m
7.Calculate the distance between the top of the tower, H, to point A.  m

  In the shaded region shown below, the radius of the smaller arc ieqg7h3a6)q+uk u29c vtqvz 4rs 66 cm and the distance between the two arcs is4 2gr3a+vctu q q9)qz7u vh6ek 44 cm.

1.Find the perimeter of the shaded region.  cm
2.Find the area of the shaded region.  $cm^2$

  Consider the shape shown below (qm.): 0*girj fst n6bvl+ 9bp+nj3nnu shaded region). The radius of the longer arc is 12 cm ant u 0nbi)v*+j s9nn:bg+l 6rq3fmjn. pd the distance between the arcs is 3 cm.


1.Find the perimeter of the shape.  cm
2.Find the area of the shape.  $cm^2$

  ABC is a triangle of a;y.rqc7 gm.z,:hmw rlrea $32 cm^2$ . The sides AB and BC have lengths 7 cm and 12 cm respectively. Find the two possible lengths of the side AC, giving your answers correct to 3 significant figures.AC$\approx$  cm and   cm

  Paper party cups in the shape of a con0f 8np v2a3wx k8lott5e are formed from a sector of paper material. The diagram below left shows the finita3o8 pnfvl2xwt08k 5shed paper cup. The diagram below right shows the original paper sector used to create the cup. When the cones are formed, the straight edges of the sector join without any overlap.


1.Find the slant height of the cups, L.  cm
2.Find the arc length of the sector.  cm
3.Find the angle $\theta$ in radians.  radians

  BT represents the height of a skyscraper, with its base a*e/fxegpxh-,id9 cm4j 8) vih t B and top at T. Jack hag8fxi,e)-mc 4jh h/vdeix9*p s an apartment on the 32nd floor at C, three fifths of the way to the top of the building.


From a point A, on horizontal ground 200 m from B, the angle of elevation to C is $38^{\circ}$.
1.Calculate BC, the height of Jack's apartment above the ground.  m
2.Calculate the angle of depression from T to A.  $^{\circ}$

  A lawn sprinkler is setup in the corner of a rectangular lawn which is enclosw:8kk - k9-ytuwwnh9ked by a fence. The sprinkler has a maximum spray distance of 5 m. The lawn hakknw-k-t 8 9:ykuhw9ws side lengths 4 m and 10 m. This situation is illustrated in the diagram below, where the sprinkler is positioned in the top right corner.


1.Find the angle $\alpha$ in the diagram, leaving your answer in degrees.$\alpha$=  $^{\circ}$
2.Find the area of grass that is currently being watered by the sprinkler
in this position.  $m^2$

  A motion sensing security light is ins;qtn9q 5t + eq-van)mrtalled in the corner of a front garden. The sensor can detect any movement up to 7 meters. The garden has boundaries of two 12 m long parallel walls, ;vq qa9)tnnm tq5r+e- and the house facade and a wooden fence which are 66 m long. This creates to a rectangular area as shown in the follow diagram, where the motion light is positioned in the top right corner.


1.Find the angle \alphaα in the diagram, leaving your answer in degrees.  $^{\circ}$
2.Find the area of the front garden that is currently being monitored by the motion sensing system.  $m^2$

  Bus-1 and Bus-2 leave a tg*wh.0veq b s5qf40u y) ykcc+erminal, T, at the same time. Bus-1 travels 65 km on a true bearin.0)h5ey*b swyvg+0 qqfuck4cg of $140^{\circ}$ to point U, then a further 40 km on a true bearing $235^{\circ}$ to point V. Bus-2 travels directly from the terminal to point V.


1.Find the distance, d, travelled by Bus-2.  km
2.Find the true bearing on which Bus-2 travels.  $^{\circ}$

Bus-1 averages a speed of 55 km/h and Bus-2 averages a speed of 36 km/h

3.Determine which bus arrives at point V first. Justify your answer. ----待选择----Bus-1Bus-2 

  A camping tent in the shape of appr o 6fcngt74f/ i0,t cone is constructed from a large circular sheet of polf60 cpn7rtpo fi/,g4tyester with a sector cut out, as illustrated in the diagram below. The circular polyester sheet has a centre O and radius rr. The area of sector cut-out, OABC, is $\piπ m^2$ and the length of the arc ABC is $\dfrac {2\pi} 3m$.

1.Determine the slant height of the tent, r.  m
2.Determine the angle of the sector OABC,$\theta$, leaving your answer in radians with exact values.

  The side lengths of a quadrilateral ABCD are AB =6 mm, BC =5 mm, CD pv 99sg zz 6fzb 4o(8ip xbz)*kop(1qk=8 mm, and DA = 9 mm. Thesikobz6g 99(vo(pf1 kpx 8pz4* z)bqz angle $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$ is equal to $90^{\circ}$.


1.Calculate the length of AC.  mm
2.Calculate the size of angle $\mathrm{A}\hat{\mathrm{D}}\mathrm{C}$.  $^{\circ}$
3.Calculate the area of triangle ACD.  $mm^2$
4.Calculate the area of quadrilateral ABCD.  $mm^2$

A point P is chosen on the side CD in a such a way that the area of triangle ACP is equal to the area of the triangle ABC.

5.Calculate the length of CP.  mm
6.Calculate the length of AP.  mm

  A picnic area is shaped like a quadrilateral, ABC eb04,zpud+dpf 1-;cc qqkzh*D, where AB = 10 m andCD = 17.5 m. Three of the internal angles have been measured; anpcp qz0+f zek* q4bdh-1c;ud,gle $\mathrm{A}\hat{\mathrm{D}}\mathrm{C}=51^{\circ}$, angle $\mathrm{C}\hat{\mathrm{A}}\mathrm{D}=70^{\circ}$ and angle $\mathrm{A}\hat{\mathrm{C}}\mathrm{B}= 42^{\circ}$. This information is represented in the following diagram.



1.Calculate the distance AC.  m
2.Calculate the angle $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$.  $^{\circ}$

There is a lamp post at A, perpendicular to the ground. The angle of elevation to the top of the lamp post from B is $32^{\circ}$.

3.Calculate the height of the lamp post.  m

Patrick's family is having a barbecue party on the picnic area. The family brings a 1.5 litre bottle of lemonade and cone-shaped paper cups.
Each cup has a vertical height of 88 cm and the top of the cup has a diameter of 5 cm.

4.Calculate the volume of one paper cup.  $cm^3$
5.Calculate the maximum number of cups that can be completely filled with the 1.5 litre bottle of lemonade.  completely filled cups

  The diagram below shows a triangular backyard26 obtugi pw0/r/y.xhrh +d7u ABC. A dog is on a leash attached to a pole at point A. The length of the leash is 10 metr g/pr07 uyw6x hhui/.ro2b d+tes.

1.Find the area of the backyard that the dog cannot reach.  $m^2$

The owner of the dog estimates that the area the dog cannot reach is $50 m^2$.

2.Calculate the percentage error in the dog owner's estimate.  %

  Two drones are being flown in a park. Drone A starts at5;vv()+yt2a i:gcnl zbe 4xrj point P and travels towards point Q in a srcl;ig b:)2(xntzy+ve45 jv atraight line on a true bearing of $115^{\circ}$ at a speed of $2.5ms^{-1}$. The distance from P to Q is 800m.
Drone B starts at point Q and travels towards point R in a straight line on a true bearing of $215^{\circ}$ at a speed of $7.5ms^{-1}$ .
These two paths are shown on the diagram below.



1.Find the acute angle $\mathrm{P}\hat{\mathrm{Q}}\mathrm{R}$.  $^{\circ}$
2.Find the distance between Drone A and Drone B after 4 minutes, rounding your answer to the nearest metre.  m

  Consider the analogue clock below, which hawoedq7 7un -6ms a circular face with centre at point O. The minute hand of the clock, OP, has a length of 12 cm and the hour hand, OQ, has a length ofn dwm6q7o- u7e 10 cm.


In Figure:1 the time on the clock is 7:00,pm.
1.For Figure: 1, find
1.the size of the reflex angle,$\mathrm{P}\hat{\mathrm{O}}\mathrm{Q}$. , in degrees.  $^{\circ}$
2.the distance between the ends of the hour and minute hands.  cm

In Figure: 2, the time is now 7:18pm, and as such, the end point of the minute hand has rotated through an angle,$\theta$, covering an arc length of $l$.

2.For figure: 2, find
1.the size of the angle $\theta$ in degrees.  $^{\circ}$
2.the arc length $l$.  cm
3.the area of the shaded sector.  $cm^2$

Another circular analogue clock, shown below, has a face radius of 14 cm, and minute and hour hands of length 14 cm. The current time shown on the clock is 6:00\,pm. The bottom of the clock face is 4cm above the base of the frame holding the clock.


The height, hh, of the end point of the minute hand above the base of the frame is modelled by the function
$h(\theta)=14\cos \theta + 18h$
where $\theta$ is the angle rotated by the minute hand after 6:00pm.

3.Find the value of h when $\theta =170^{\circ}$.  cm

The height, gg, of the end point of the hour hand above the base of the frame is modelled by the function

$g(\theta)=-14 \cos (\frac{\theta}{12}) + 18$

where $\theta$ is the angle rotated by the minute hand after 6:00pm.
The end points of the minute and hour hands have the same height from the base of the frame for the first time when $\theta=k$.

4.Find the value of k.  $^{\circ}$

  An airplane leaves Doha airpo e9hx/a 6jfqhu*ga p.fh-4 hx8rt bound for Paris Charles de Gaulle airport. There are lights located at the end of the runway at points A(0, 4hajf fe96p*-hq8uxga hx4. h/) and B(6, 8), relative to a terminal at the origin. The takeoff path of the airplane is the perpendicular bisector of line AB.

1.Find the equation of the takeoff path in the form $ax + by + d = 0$, where $a, b, d \in \mathbb{Z}$.

The airplane travels at an average speed of $570km\:hr^{-1}$ in a straight line. Once the airplane has reached cruising altitutde, an air traffic controller at the top of a 200m high air traffic control tower at C(7,0) observes that the angle of elevation to the airplane is $40^{\circ}$. Five minutes later, the controller observes that the angle of elevation is $10^{\circ}$.

2.Find the cruising altitude of the airplane in metres.  m

As the airplane is about to land at the Paris Charles de Gaulle airport, the pilot is asked to delay the landing due to a traffic issue. The pilot is instructed to turn the airplane on a bearing of $045^{\circ}$ for 10km until reaching point P, then travel on a bearing of $165^{\circ}$ for 30km to point Q before flying back to the original point O for landing.

3.Find the angle OP̂Q.  $^{\circ}$
4.Find the shortest distance from Q back to O for landing.  km
5.Find the bearing the airplane must travel on to get back to O from Q.  $^{\circ}$

  The temperature in New York ra 9vwt, ha-od:-rqg lk(nges from $-4^{\circ}C$ to $22^{\circ}C$ throughout the course of a year. The temperature, T, can be modelled by the function
$T(t) = -a\cos(bt)+d$,
Where t is time measured in months from the beginning of the year, and a, b and $d \in \mathbb{Z}^{+}$.

1.Show that
1.a=13;
2.b=30;
3.d=9.
2.Sketch the function T(t) for $0 \leq t \leq 12$, clearly labelling the coordinates of the maximum and minimum points.
3.Find the temperature when t=4.  $^{\circ}C$
4. 1.Determine the number of months in a year that the temperature is above $9^{\circ}C$.
2.Find the probability that on a randomly chosen day of the year, the temperature is above $9^{\circ}C$.
5.If the temperature in New York ranged from $-7^{\circ}C$ to $25^{\circ}C$ instead of $-4^{\circ}C$ to $22^{\circ}C$, describe how this would affect the probability found in (d) part (ii).

  A large plot of land is/ b pvh,rbj)+a shaped like a trapezoid, ABCD, where AB is parallel to CD, AB = 1rabp, /h bj+v)2 km and CD =8 km. The diagonal BD of the trapezoid is equal to 10 km. The internal angle $\mathrm{A}\hat{\mathrm{B}}\mathrm{D}$ is equal to $58^{\circ}$.

Let a new point, H, be a point on AB closest to D.

1.Calculate the distance from H to D.  km
2.Calculate the length of AD.  km
3.Calculate the size of the angle $\mathrm{D}\hat{\mathrm{A}}\mathrm{B}$.  $^{\circ}$

A landowner estimates that the length of AD is equal to 11.5 km.

4.Calculate the percentage error in the landowner's estimate.  %

The landowner decides to install cone-shaped concrete bollards along the perimeter of the plot of land. The bollards are to be installed at a distance of 100 m from each other. The radius of the base of each bollard is 20 cm and the height of each bollard is 40 cm.

5.Calculate the volume of one of the bollards.  $cm^3$
6.Calculate the total volume of concrete needed to install all the bollards.  $cm^3$

  The quadrilateral ABCD represents a recrvc xasf-yhj9 m do:27 bk+q/*ieational park, where AB =250 m, BC =110 m and CD =130 m.da2/9is7vo+ - :mb*khq xjfyc Angle $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$ is $80^{\circ}$ and angle $\mathrm{C}\hat{\mathrm{D}}\mathrm{A}$ is $115^{\circ}$. This information is shown in the following diagram.


A straight path through the park joins the points A and C.

1.Find the length of the path AC.  m
2.Show that the angle $\mathrm{D}\hat{\mathrm{A}}\mathrm{C}$ is $27.5^{\circ}$, correct to three significant figures.  $^{\circ}$
3.Find the total area of the recreational park.  $m^2$

A new path, DE, is to be built such that E is the point on AC closest to D.

4.Find the length of the path DE.  m

The section of the park represented by triangle CDE will be used for a school cross country carnival. A track will be marked along the sides of this section.

5.Calculate the total length of the track CDE.  m

  The diagram below showsrt mfkl37g) :a a triangular park ABC. Alice is in the park with her dog. She stops at the corner B of the park to make a phone cr m:)7 kfga3tlall. The dog's leash is 12 metres long.


1.Find the area of the park that the dog cannot reach when attached to its leash.  $m^2$
2.Determine the percentage of total park area that the dog can reach while attached to its leash  %

  On the diagram below, points X and Y aresaaf8s;5rk1 l9d k;hrmg;p l: the centres of two circles containing tg l 5hk l far;msp1:k;a98sd;rhe two arcs from A to B.

1.Find the length from point A to point Y.  mm
2.Find the perimeter of the shaded region.  mm
3.Find the area of the shaded region.  $mm^2$