Mary (M) sits on the top of the Eastern side of the Grand 6bami4:siy m ez33 yw )sk0 mdq(s.5ogCanyon. Below her stands Peter (P), at the bottom of the canyon. It is known that the depth of the Grand Canyon is 1850 m ek5g4 dos)sqabi(0m3:6w. yzmims3 yeters at this 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 le2x-2pe-ro, pxr q0e zn3hq/r: ;osp hnans against a vertical wall. The distance from the top of the ladder to the ground is 3.1hoqr neh,q 3px-ex-nz2p pr :0 2;ors/ 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 triangle ABC,1w/*oqagc qj1quy6 *jnv-pv. 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 a*aorh r1 i).p n+ y3jtjqw3z)hm/ls3qn 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 playgrou:qc-n5 ru axl6je) vh/ 2*vpuynd, the grass is watered by lawn sprinklers that are set to spravr -6 qphu/j:5axeu2)v*lc nyy 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 yn bb qn(hvhoh*t.50,2a6p e uthe blade is in motion, it mob qht0(ne5*hv ,. y6au phbn2oves 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 below a 120 m high skyscraper.c)*9k 1(y8fjrtnh0 i4zd xfwh.7ht sh From the top of the skyscraper, the78r*.twf yc hhzh)14tnd xs(i9jfk0 h 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 farming land is shown in ty( 0m+p h7sj *eyfwhz8he diagram below. The longest side is 120120 m, with the twomsy yhp(fh wz*8+7ej0 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 di. y j8)x9wm1,dk(km bnbvr0snagram. Side jm ywn0mr1n9x., bkkv(db s)8length AB 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 1.8 m and r- 7d6vgp7h2 uh8-du)nbclw9hxe(m wpadius 9.2 cm. A wedge is cut from one log and the cross-section n8 )9 h cwwpuh dp-dhe(7lg2v-bum67xof this 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 shown in the folll1f oy4942+w aj a4uff nrgh+iowing diagram. Side length AB is 150 m and side lengtfy+4 +92n a4wff g1iajhrl 4uoh 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 tripo:l: 8p 9wqwdb, Jim stopped and checked his compass for directions. The compass has a circq8b:w od9wl p:ular face with centre at O and the 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 waters part of a lawn, shown as the sha+h()j jqvu m(ew t(tz9ded area in the diagram bjw) (tet(jm 9zuq+hv(elow. The sprinkler covers a distance of 13m, reaching a maximum distance of 17m 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 of a warehouse, shoo8nm sv xx6s++mo j4;own as the shaded area in the following 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 anvmms 4 6osoj++ox8xn;gle 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 radiks,ul9 uf8l 8hfiv/ (vus 7.5 cm are used to build a wooden cabin. A wedge is cut from one log and the cross-section of this log is illustrated in thkvfl 9us ih8,lu(v f8/e 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 .u trpr o/0iz.ABCD. The vertical height of the pyramid, VM, is 5 cm. The sloping edges a / zr.ip0u.torre 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 AC=4 u5ayzkvb .+7hffqm7r :4fb1 scm. The sloping e:a + v1u.7bqz4yfh ffs b5k7rmdges 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 island located off the nort 3ub9i//ef,iiz1 y tiwh-western tip of British Columbia, Canada. Two side lengths of Triangle Island are equal to 1600 and 1800 metres and the angle between these two sides is ezb iw /1y,iiut/fi9 e3qual 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,yqslp cld sl(6l13p9/ and CD = 11cm. The size of angle cpdq p3y(ssl 9/ll1 6l$\mathrm{A}\hat{\mathrm{D}}\mathrm{C}$ 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 m, DC p;ck/kj5ru t5= 7 m and BC = 55t;rupc/k j5k 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 2.2 m*duv08r i (q+ 1i zv2c+zxxibw0kmfmav 1 y52h and radius 7.6 cm are used to construct a fence line. Part of the logs are remozi020i1ryxqu+v wv( b8h i2 + xdfmc5m*a v1zkved, as illustrated in the diagram below.

Find the volume of each log.  $cm^3$

  Consider the analogue clock below. The cqy5myw akvb9ulw q u 26x+9;+rlock has a diameter of 30 cm and is shaded bxwyymqu+rqbkl5va+99 u; w6 2 etween 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 speciality birth8522mg2z4 mj -xiey4hwujgp kday cakes. The box has a rectangular prism base, with a rectangular pyramid top, as illustraze mu-xwkg4mi2g85p 4 hj22 yjted 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 if;h 5a +ut;txqn his backyard to play with his children. The field is cover;q5 xf+t athu;ed 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 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 creating colf: jtp1q5i. wn1ml8ene-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 alw .jmn85tlq 1ie:1fp nd . 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, withc- xnt1k5ml gg,w5vwq,wm45w AB = 3 cm. The vertical height of the pyramid, VM, is 12 cm.,vn5wktgqc-m4wl , 5 wgxw15m


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

  The sides of a triangle ABCv-dhd5cjs2 u - have the following lengths AB = 40 mm, BC = 30 mm, and Cvd-25u-sc j hdA = 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 xu.49wgytwr wi+( nn/the sky 50 m apart from each other on the same height above a flat (horizontal) ground. Two straight ropes, AP and BP, are attagtyw.iu9 (wx4r +/w nnched 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 p 64 p0 uh8 d*-chboqh7ib.ehqqlot has a curved border which can be described as an arc, PQ, of a circle qbqh.*p h ui dhbc7eq0 h68-o4with 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 gardeblxkl*i9e s95n, ABC, is shown in the diagram below. A walking path pasb*lx es5k9 il9ses through the 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 circles with centre O are shown in the diagram w 873we8nbq octk/ w,or( v7zbbelow. 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 circumference of the larger circle such thv/krowtw woe7b,q n87z(8bc 3at 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 a4kyhd0y1gxj 3 nd $\mathrm{A}\hat{\mathrm{C}}\mathrm{B} = \dfrac{\pi}{6}A$.
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 cmeuncdoha v,* p);w9c3 . 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 lengths a = 10.2 cm, b= 17.5 cttu3uj4qx 8 tj;r(4s;ki2 d yim 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 d5q8d nea*f8bin the shape of a right-angled isosceles triangle, ABC.AB = 6 5be a*n8dd8fqcm and angle $\textrm{A}\hat{\textrm{C}}\textrm{B}=90^{\circ}$.


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(vlg*s947u l a 0wcfbj, the angle of elevation to the top of a 20 m high clif clba vjgfsw(lu40*79f 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 complexd,balecc;ennns7-r-- 3mz g/ , where AB = 210 m, Bb,z37sd - gnacclmnr- ;n/e-eC = 90 m 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 is o3;4v ug lw25 ac286w bwbryoishown below, where AB = 56m, AD =122 m, CD =40 m, and the anglescw5i 4r2g3 8y a2wbv;b6o uowl $\mathrm{B}\hat{\mathrm{A}}\mathrm{D}$ 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 afe j-l-t9.hy, kdkk) d building. The angle of elevation from a point on the ground 40m from the base of the building to the- -,)htdkjl. ek dkf9y 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 has sidp ,o4njt9g u0ie lengths AB = 25 cm and BC = 35 cm. Thejo0i4t9pgu n , angle adjacent 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 cnv.o5gb1 7cv7d ns1ef q+8jh from a lookout at the top of a lighthouse (A) down to a boat locatehv1c b1v j7o7.58ns nge+dcqfd at point 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 sett 7jo, u2iqv:ca:it2w cing up the tent, she goes for a walk to see the nearby surroundings. She walov7w cc:t au2,q2:ji iks 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 hwrb:lgv7qtg)+ey48 e pg4jo5orse trek through the Saharan desert. They start from a ho5r 4gte8w) p q+gjo4eg l:b7vyrse farm, 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 si7 jf3jt1xwig g6y4:ppdesAB=10 mm and AC=8 mm. Given that the area of the triangle i3 i7w1 g ytgpp6:x4jjfs $24 mm^2$, find the possible values for the length of side BC.BC$\approx$  mm and  mm

  Daniela (D) stands nmi55sp4vhcq/q j /u4qear the window on the second floor of the six-story apartment building and can see Andrea (A), who is 14 m from the base of thqq/hj u m/q54i5vcs p4e building. The angle of depression from Daniela 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 below. A r5bs adlanfi87biw1m 9ll/ u /5iver 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 alif n/ds5a /bu15i9w 8l bam7lngle $\mathrm{A}\hat{\mathrm{C}}\mathrm{B}$ 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 smallgwix) 7u1 vz86 lz+maier arc is 66 cm and the distance between the two arcs is 4i v7zazu im1glw8 6x+)4 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 radius of the ly hf7lwabx -.zne o9/6onger arc is 12 cm and the distance betxa 97lo.f6 hw-yb/zneween 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 ayk;e8g0p 3;thh i0c bq3 go- r/wjx:ukrea $32 cm^2$ . 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 cone are formed from a -5 vt(-/akw osaxi f;lsector 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 straight edges ol-;twa5fksi o /v(-axf 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, w2 x -i-stbqq4z6xyc/ *xl:ine ith its base at B and top at T. Jack has an apartment on the 32nd floor at C, three fif xby/-ii xqx6cnltq4-2ze s:* ths 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 encloxn i*g0y0fs3dsed by a fence. The sprinkler has a maximum spray distanc *f yi0dx3sng0e 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 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 installed in the corner of a fro-mt6r0aha yo8*7 uj)bfv,w sxnt 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 fence which are 66 m long. This creates to a rectangular area as shown in the follow diagram, w78w -uja ovfxba ym*0st rh6),here 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 a- qqw: ig-s*xterminal, T, at the same time. Bus-1 travels 65 km on a true bearxg* qsa:wq i--ing 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 ofskj;je)5sbou uz3b ;zp4.lf4 a cone is constructed from a large circular sheet of polyester with a sector cut out, as illustrated in the diagram below. The c.4 zssuklbe;ub4pj3 )ofj5z; ircular 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, BCdns+ltyf8y-cj ,5 (2 usen f9v fh3e.n =5 mm, CD =8 mm, and DA = 9 mm. The angl,ljdu3s .yf f2v+y cn(f59ehnn-est 8e $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$ 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, ABCD, where AB = 10 m a)8m kvzdlj 8s:ndCD = 17.5 m. Three of the internal angles have mjldz88s):vk 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 ABCh1,zv)gz9lk)y 2cox h. A dog is on a leash attached to a pole at poic yh,2zhz og)xv9lk )1nt 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.70t*w bc)y2o zpn(cyv-tt4 po Drone A starts at point P and travels towards point Qy( vob) 4pncy7* tot- 0ptc2wz in a straight 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 has a circular face with centrkstoq f2l1kixo dr ;g9*l1q43e at point O. The minute hand of thekqor qf2 * ti139o4klg1xd;l s clock, OP, has a length of 12 cm and the hour hand, OQ, has 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 Doha airport bound for Paris Charles de Gaulle air1a0 nraw n, oi/h uksa4:)jcmk1 h.f8nport. 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 tak an4r,hsn)1w.oh8 /ckk1a:0 n imujafeoff 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 Yor-* u0yyc0yaqu3ib 3w vk ranges 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, where AB is parallel 7xji3rk;+ bqbjze/2 ito CD, AB = 12 km and CD =8 km. The diagonal BD of the trapezoid is equal to 10 km. The internk 2 jbzier/b;73qix+jal 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 ABCv;qx-vz zdt b:m2b)pq 1 fv9a:D represents a recreational park, where AB =250 m, BC =110 9 bz x2vfp1dabq ;)tv:qm: z-vm and CD =130 m. 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 shows a triangular park ABC. A;zmwyl9m*c ugk hjq7mm8;l o zvf9.6xh:ft8, lice is in the park with her dog. She stops at the corner B of the park to make a phone call;wfzho g: y8z6lf9 qm*k;.l,9j m mc v87huxtm. 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, psbh3+:w cep2p il n x2tsbdor;n18x.1 oints X and Y are the centres of two circles containing the two arcs from A to B. tcp18 l: ndnx2e; .wx3sr2b1s +bioph

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.  $mm^2$