Mary (M) sits on the top of the Eastern side of the Grand Canyon. Below her stam8*mw02y gyz na/q 3phnds Peter (P), at the bottom of the canyon. It is known that the depth of the Grand Canyon is 1850 meters at this point. Nathan (N) stands on the Western side of the bottom of the Canyonnma 32*w ypyhzq m8/g0, 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 disqb2 *i+-yzkzd tance from the top of the ladder to the ground is yzqz2ik*-bd+ 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 a trianmzs,xa0ba4p-y2*:;y vc 6l gd4yq8o6v hvsg r gle 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 watering a garden sprays at an oxdo9uvj+ 73w hv/xe a*l 9c26u kk5inarc 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 lawnfud,8t- w2d u5jx 4jln sprinklers that are set to spud5jl-d4 f,wn28x ujtray 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 blade has a length of 0.45 m. When it's raining and the blaa6hrn cmia 1.t 1ow1guyx+/v0de is in motion, it moa1/ar xm0hno+cw6gty1i 1v .uves through an arc 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 belo3e fwe+s 89ecqw a 120 m high skyscraper. From the top of the skyscrae3feswe qc9+8per, the traffic lights can be 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 far8gf+ scsv:0yeming land is shown in the diagram below. The longest side is 120120 m, wgces v80+yf:sith 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 es z7 oa;5srlo 320evca triangle, ABC, as shown in the following diagram. Side length AB is 49 km and side length Bz2;vooal0s3s r 5ce7 eC 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 cylindricaajioryzck,0:(q6k5 h l logs of length 1.8 m and radius 9.2 cm. A wedge is cut from one log and the cross-section of thishrzcja :,ok(yiq50 6 k log is 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 triangle, ABC, as sc c2d;/ gkk psr/6s//+qq hkpwhown in the following diagram. Side 6dh2 p /qrqk/cp kkgsc;/+sw/length AB is 150 m and side length BC is 200 m. 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 compass for directions.bow) u v1(n8+kjx rqd*a97hb a The compass has a circular face with centre at O and th(baw+9 rbnxu) d78 vj1*aok qhe diameter of 20 cm. The needle of the compass is at an angle 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 watm cfj p88, 3 hosm(*ldvw1kmz4ers part of a lawn, shown as the shaded area in the diagram below. The sprinkler covers a distance of 13m, reaching a maximum distance of 1*d(pf88 wcj1hks4ommv ,lm 3z7m and covering 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 p 0r8., bmb)pxij(ehjof a warehouse, shown as the shaded area in the followinehb mp,)bpi. 0jxjr8( g 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 cm are us5qbt5 2i5ll (emo/5vt+pv uebed to build a wooden cabin. A wedge is cut from one log and the cross-section of this log is illustrated in b lpem5quvo(55l+ i/ tteb25vthe following 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 square base ABCD. The vertical height of the d+x.p6src 64+ tyfl oopyramid, VM, is 5 cm. The x o+s l6+yf.po6r 4tcdsloping 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 witxc( gnb 3(8asoiv6t2e1* dbdlh the base diagonal AC=4cm. The s8g dts xcebb6lanov2((1 3*d iloping edges are 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 ie-54l: wfv)rx.vceb 8x))bjq otw 7cusland located off the north-western tip of British Columbia, Canada. Two side lengths of Triangle Island are equal to 1600 and 1800 metres and the angle xbe w8. q cf:v7lbtox)- wu vre)j4c5)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 i/ppz.efll *j9d c3 c.=17 cm, BC = 12cm, and CD = 11cm. The size of a*3.l9j.cldiz e fpc/pngle $\mathrm{A}\hat{\mathrm{D}}\mathrm{C}$ 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 mg1fj) )q8.l tddf lnp2, DC = 7 m and BC 8dd)nf 2 pt g)qlljf1.= 5 m. 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 2.2q , 1chjvz- dak7rupj- x;2wa *u2:yze m and radius 7.6 cm are used to construct a fence line. Part of the logs are removed, as illustrated * kw7 xr,2uhy:p u-2je;djvzzcqa - a1in the diagram below.

Find the volume of each log.  $cm^3$

  Consider the analogue clock below. The clock has asxtkf/f-z* b+ diameter of 30 cm and is shaded +f s/*fzxtkb -between 1 and 6.


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 specina;udh htj2mfb fr9ep7 p ig0ef o(d:67- 36acality birthday cakes. The box has a rectangular prism base, with a rectangular pyramid top, as illustrat 7;9eh7 :pfgbj-ddi(aa6 etuf6o23 cmp0nhf r ed in the diagram below with dimensions 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 po8vqnujckjbm5q;b* j p 6 t-su+8w t12wq(9fhlay with his children. The field is covered by grass in the shape of a sector of o+8jjqp(2w5-k q fhunq9*vt wbt b8ucm61 j;sa circle with radius 8 metres, as shown in the following diagram. The angle at the centre 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 creati4 f2mrzz jgcl+ou . tb;3pl4s;ng 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 the diagram below. The paper circles have centre O and . The length of arc ABC is pc ; ;rlgl bt z4uz3jm+fo.2s4$4\pi$ 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, with AB = 3 cm. The vertica :n14l rp-mchxrbe 1,g pp;o.tl height of the pyramid, VM, is 12 cxgpbt-o;ce,:1m lp4 n rh.1prm.


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 = 40 mm, BC = 30 mm, 6kv4w2- xzr dkg jy.r;and CA = 55 wygk dkzr6-xrj;v2.4mm.



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 balloonswkdbxa 7u kgw*cb5 (m30f9m)f:4 ani h, located at points A and B, float in the sky 50 m apart from each other on the same height above a flat (horizontal) ground. Two straight ropes, AP and BP, are attached to the same point, P, on the ground. The length of AP is aam 49wfc gi whk0n7k3f*bx) (u5:mdb
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 9.uvapo r)vk1wqy+w* below. The plot has a curved border which can be described .pwkv) a1vuq rw9y+o*as an arc, PQ, of a circle 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 shown in the diagram below. A walking path paz/a7ct/s . 1zaz5ubkj sses through the17zu/5tazc b/ .szj ka garden at point D and leaves at point E, such that DE is parallel 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 circleyqxk.xf4/ti8 b.j*y es with centre O are shown in the diagram below. The radius of the smaller circle is 5 cm and the radius of the larger circle is 7 cm. Points A and B lie on the circumfey b*q .8i/yxtek4fj. xrence 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:;zio w 8xsw)g; rubqc*em/. z=3 cm,BC=5 cm and $\mathrm{A}\hat{\mathrm{C}}\mathrm{B} = \dfrac{\pi}{6}A$.
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.g(x a(,tu8jh/voj; y:m,a zni3 9-srj 2rubtw The area of the 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 f7-h . oeqlfs1lengths 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, A t rvct owhhrg0z)jmv91v033+BC.AB = 6 cm and actwvorhv9 )3jz3m 0tg+hr 0v1ngle $\textrm{A}\hat{\textrm{C}}\textrm{B}=90^{\circ}$.


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 angle opp:iueo+amx(k.dsr f ;60 3rf f elevation to the top of a 20 m high .;pp:dsoar+ ief 3kf(u0xm 6rcliff 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 compt(;+v,w*encp osqt4kh3e. t7x j3)aa ki i h2mlex, where AB = 210 m, BC = 90 m ve,3a.h4 3ct)2s*hqk mi+ jti;7wp(nxokae t and CD = 130 m. 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 *z3v9vtzq/ oy is shown below, where AB = 56m, AD =122 m, CD =40 m, and the ang 3*zq/vy9vzot les $\mathrm{B}\hat{\mathrm{A}}\mathrm{D}$ 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 buildinba*fzg styi91d. p92cg. The angle of elevation from a point on the ground 40m from the base of the building to the bottom of the fla9 zyd 9cp2t1.giasf*bgpole 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 has side lengths AB qr6wj rx/* .v hcv,w,b= 25 cm and BC = 35 cm. The angle adjacentvx,ww*jv /r hc6r,.bq 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 lookout at the topte x2d. kr:.of of a lighthouse (A) down to a boat located at pointf td.ox2:r .ke 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 settqoz :gqv-.yp;t :s4i ping up the tent, she goes;p4-: vqtzpy.i soq:g for a walk to 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 Moroccotr 3rq qgd)oi;n og3/*, a group of tourists experience a horse trek through the Saharan desert. They start from a horse farm, moving on a bearing ooo;r*3gq)tiqn d/3gr f $075^{\circ}$ 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 mm and AC=8 mm. Givan7np mk0z h/,en that the area of the tr70pmz /knh na,iangle is $24 mm^2$, find the possible values for the length of side BC.BC$\approx$  mm and  mm

  Daniela (D) stands near the window on the second floor of the six-story apartmentw m6-n3 p1rx5+ul qlvt building and can see Andrea (A), who is 14 m from the base of the building. The angle of depression from Daniela to Andrea is rmp3x-tv + nwq65ll1u $20^{\circ}$.


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 below. A river runs along the ;z *(ki dcl*kfayc 8iq1y) -ccw iby8/edge from A to B and the farm house is located at C, a safe distance )q y8i*k;ci fck*y 8laicbwd/zcy -(1from the river. The distance from the farm house to point A is 270 m and to point B is 360 m. The angle $\mathrm{A}\hat{\mathrm{C}}\mathrm{B}$ 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 /w6c l61rddih0 acv1f of the smaller arc is 66 cm and the distance betweena0vf6l 1i/d drcch6 1w the two arcs is 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 (shaded region). The radipfa 5/bxreutb)6.b q.us of the longer arc is 12 cm p6ufaq.b/5bt). rb x eand 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 ofv;l+gqx (xc6h area $32 cm^2$ . 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 t j0gvpr2qon)fa mt, y:gj::9 t:yi x1yhe shape of a cone are formed from a sector of paper material. The diagram below left shows the finished paper cup. The diagram below right shows the original paper sector used to create the cup. When the cones are formed, the straight1 9 ti:opn0m j:jfyaq: y2),grxg ytv: 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 2-a 0jimrik2d its base at B and top at T. Jack has an apartment on the 32nd floork2-jdma i02 ri 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 fd -b5a;/u,;smkrkwgcac4 n,lx /t9 l enclosed by a fence. The sprinkler has a maximum spray distance of 5 m. The lawn has side lengths 4 m and 10 m. This situation is illustrated in the diagram below, where the sprinkler is positione- llta,nc u95 wkfx;k cabg/;4,mrsd/d 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 installed in the corner of a front-s ezd,kf6a )3v4xq a*-nnf+vbfs . pw garden. The sensor can detect any movement up to 7 meters. The garden has boundaries of two 12 m long parallel walls, and the house facade and a wooden e6v4 nf ) qpv*.wffx,3 ksa-bas+zd-nfence 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 terminwt4 d numo(,8jal, T, at the same time. Bus-1 travels 65 km on a true ,ujt nodm 84(wbearing 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 a cone is *b4 l0djynj33 2xkpwgiws 8-b4q( eibconstructed from a large circular sheet of polyester with a sector cut out, as illustrated in the diagrams3 ejlp jwdq0x4- ii 3w*(2 kbbb4gy8n 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,z +g,)y p5paju68na1x ,px5:o lyw bun CD =8 mm, and Dx pja5xbn,y w ,6uo p8yazul g5p)+:1nA = 9 mm. The 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 shapm f0xoi(pt bf c)2lohc,8jn /lqf*)k( ed like a quadrilateral, ABCD, where AB = 10 m andCD = 17.5 m. Three of ni2mqtfb/8x)o(oflc k)0,c pjhf( l*the internal angles have been measured; angle $\mathrm{A}\hat{\mathrm{D}}\mathrm{C}=51^{\circ}$, 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 backyard ABC. A dog is on a lec7r:expa2a; o ash attached to a pole at poic xe ao7ap2r;:nt A. The length of the leash is 10 metres.

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 star( kp:n6r/z xjnts at point P and travels towards point Q in a straight line on a true b:(pkrj n6n xz/earing 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 has a circular face with fy9l2h*e; rmtcentre at point O. The minute hand of the clock, OP, has a length of 12 cm and the hour hand, OQ, hasf l9e tymr*;h2 a length of 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 Dohadu oe4 ;n7+/w:ovni(j pqwou7 airport bound for Paris Charles de Gaulle airport. There are lights located at the end of the runway at points A(0, 4) and B(6, 8), relative to a terminal at the origin. The takeoff path of the airplane is the perpendicular bise7(ovodq4w7 jepi+ /nwuonu:; ctor 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 raz) te opm(ezyv(0k/.+oflu*/rg rjn 1 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 shaped like a trapezoid, ABCD, s*etv 50v*q zb9m ol8iwhere AB is parallel to CD, AB = 12 km and CD =8 km. The diagonal BD of the trapezoid is equal to 10 km. The inteb*sei*t lm5vz q8vo 90rnal 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 recreational park, where ml3,f4ktjf cr1/aw: cAB =250 m, BC =110 m and CD =130 m. Af,3 rca /m 4fjwt1clk:ngle $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$ 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 shows a triangular park ABC. Alice is in the park w.rdy)fe6 xrw4z3xr:n gf p.4kb+ m7wnith her dog. She stops at the corner B of the park to make a phone call. The dog'sw)mnd6kyf zr w3e4 bx+.rrnp47 xg.:f 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 are the centresgep;*3 h(e.p(nz rvwgt9j h r2uj9w (j of two circles containing the two arc *te29v;juzj(.w r jg3 h(9pwghe(np rs 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.  $mm^2$