Assorted Geometry
A search and rescue helicopter shines a light down from a vertical height of 50 metres as shown
below. The circular area of light it creates on the ground has a diameter of 10 metres.
Question 1
The helicopter is elevated an additional 15 metres away from the ground. The diameter of the circular area of light on the ground is now closest to
A. 8 m
B. 13 m
C. 15 m
D. 20 m
E. 25 m
Question 2
The helicopter moves to a height so that the diameter of the circular area increases from 10 metres to 40 metres.
The area of the circular light is now
A. four times what it was before.
B. eight times what it was before.
C. sixteen times what is was before.
D. thirty-two times what is was before.
E. sixty-four times what it was before.
Question 3
The value of x in the following figure is
3.
A. 20
B. 25
C. 33
D. 45
E. 55
[Hint: Separate out the similar triangles and match up the corresponding sides and angles]
Question 4
Ben is making a 1:100 model of a car with an engine capacity of 2.3 litres (2300cm3 ). If Ben wants to include a scale model of the engine, then the capacity of the model engine should be
A. 0.0023 cm3
B. 0.023 cm3
C. 0.23 cm3
D. 2.3 cm3
E. 23 cm3
Question 5
Triangle ABC is similar to triangle AXY.
AX = 3
2 AB
If the area of ∆ABC = 108 cm2 , the area of ∆AXY is
A. 32 cm2
B. 48 cm2
C. 54 cm2
D. 72 cm2
E. 81 cm2
B
X
A C
Y
2522
10
x
Question 6
A cylindrical block of wood has a diameter of 12 cm and a height of 8 cm.
A hemisphere is removed from the top of the cylinder, 1 cm from the edge, as shown below.
The volume of the block of wood, in cubic centimetres, after the hemisphere has been removed is
closest to
A. 452
B. 606
C. 643
D. 1167
E. 1357
Question 7
A triangular prism with a cross-section of an equilateral triangle is shown on the right.
The side lengths of the triangle are 4cm and the length of the prism is 10cm.
The total surface area in square cm is
A. 46.93
B. 80
C. 93.86
D. 126.93
E. 133.86
4 cm
10 cm
Question 8
A proposed swimming pool is to be constructed in the western suburbs of Melbourne. The design of the swimming pool is shown in the diagram below. The pool has two sections: one section has a flat base, while the other section has a sloping base.
From the shallow end of the pool, the first 25 metres of the pool has a constant depth of 0.9 metres.
Halfway along the length of the pool, the depth begins to increase at a constant rate, reaching a maximum depth of 2.2 metres.
(a) Name the shape of the quadrilateral ABCF.
___________________________________________
(b) Calculate the distance EI. Write your answer in metres, correct to two decimal places. [Hint: draw out the triangle that is involved in calculating EI].
(c) Calculate the area of the side of the pool bound by ABCDEFA. Write your answer in square metres, correct to one decimal place.
(d) Using your answer to part c., find the volume of water required to fill the pool. Write your answer correct to the nearest cubic metre.
(e) The sloping section of the base of the pool bound by the rectangle BCGH is painted first and that section of the pool is filled before the flat section of the base is painted.
Calculate the volume of water required to fill the section of the pool with the sloping base, up to the level of the flat base. Write your answer correct to the nearest cubic metre.
[Hint:Visualise the section that will be filled up with water and draw the 3-D diagram that represents it]
Question 9
The top two-metre section of a five-metre high cone is removed.
Calculate the percentage of the total volume of the remaining (bottom) part
Show the relevant working even for the multiple-choice questions.
1. Given XZ = 8, XY = 10, sin
α = 3
2 , then Sin β equals
A. 4
15 B. 15
8 C. 12
5 D. 4
5 E. 6
5
2. A rectangle is 8 cm long and 6 cm wide. The acute angle θ , correct to the nearest degree is
A. 37º
B. 41º
C. 49º
D. 74º
E. 83º
3. In the figure shown (not drawn to scale), ABCD is a rectangle. The angle ACD is equal to
A. tan–1 0.1
B. tan–1 0.25
C. tan–1 0.5
D. tan–1 0.75
E. tan–1
3
4
θ 6 cm
8 cm
8
Y
Z
10
α
β
X
A B
CD
40 cm
30 cm
4. In ABC, the length AC in centimetres is determined by evaluating
A. 0
100 96 cos 120+
B. 0
100 96 cos 120−
C. 0
100 96 cos 60−
D. 64 36 96+ −
E. 0
100 (1 2 cos 120 )+
5. A yacht follows a triangle course MNP as shown.
The largest angle between any two legs of the course is closest to
A. 50º
B. 70º
C. 120º
D. 130º
E. 140º
6. For the triangle ABC, ∠ABC = θ, cos θ equals
A. 1
4
−
B. 1
2
−
C. 1
4
D. 1
2
E. 3
4
7. The correct expression for the area of the shape shown is:
A. o1 6.13 4 sin (80 )
2 × × ×
B. o1 6.13 4 cos (100 )
2 × × ×
C. o1 6.13 4 sin (100 )
2 × × ×
D. 1 6.13 4
2 × ×
E. None of the above
θ
A
B
C
3
4
2
M
N
P
5 km
6 km
10
Problem Solving Questions 8 to 11 (Copy out the diagram)
8.In the above figure, AD = 35 cm, BC = 16 cm and
a. What is the size of ? Give your answer to 1 decimal place.
b. What is the length of AC ? Give your answer to 1 decimal place.
c. Find the area of triangle ABC.
9. A flagpole AB is secured by guy wires AC and AD. The wires are secured 5m apart at points C and D. Work out the length of the flagpole to one decimal place.
A
1100 350
C
250
DB 5 m
Copy the diagram and show detailed workings in your response.
10. Children using the swing, shown below find that if they swing high enough, they will see over the fence. The swing is 0.9 metres above the ground originally.
The swing makes an angle of 62º when it moves from position A to position C.
Find the vertical distance x above the ground after the swing moves through 62º?
B
A
C
0.9 m
2.4 m
x
620 2.4 m
ground level
11. A farmer has his house built near a river. The house, H, is 780 metres from the pier, P, and 325 metres from the swimming platform, S, shown in the diagram below. Beside the river are two paddocks, PHS and FHS as shown.
(a) Find the area of each paddock.
(b) Find the length, FH, in metres, correct to one decimal place
2. The diagram below shows the location of three boats, A, B and C.Boat B is on a bearing of 110° from boat A. Boat B is also on a bearing of 035° from boat C. Boat A is due north of boat C. The angle ABC is
A. 35°
B. 65°
C. 70°
D. 75°
E. 110°
3. A cross-country race is run on a triangular course. The points A, B and C mark the corners of the course, as shown below.
The distance from A to B is 2050 m. The distance from B to C is 2250 m. The distance from A to C is 1900 m. The bearing of B from A is 140°. The bearing of C from A is closest to
A. 032°
B. 069°
C. 192°
D. 198°
E. 209°
4. The distances from a kiosk to points A and B opposite sides of a pond are found to be 12.6 m and 19.2 m respectively. The angle between the lines joining these points to the kiosk is 63o .
The distance, in m, across the pond between points A and B can be found by evaluating:
A. o1 12.6 19.2 sin (63 )
2 × × ×
B.
o
19.2 sin (63 )
12.6
×
C. 2 2
12.6 19.2+
D. 2 2 o
12.6 19.2 2 12.6 19.2 cos(63 )+ − × × ×
E. 1
( 12.6)( 19.2)( 63) , where (12.6 10.2 63)
2
s s s s s− − − = + +
5. Marcus is on the opposite side of a large lake from a horse and its stable. The stable is 150 m directly east of the horse. Marcus is on a bearing of 170° from the horse and on a bearing of 205° from the stable.
The straight-line distance, in metres, between Marcus and the horse is closest to
A. 45 m
B. 61 m
C. 95 m
D. 192 m
E. 237 m
6. A man walks 4 km due east followed by 6 km due south. The bearing he must take to return to the start is closest to:
A. 034 o
B. 056o
C. 304o
D. 326o
E. 346o
7. A boat sails at a bearing of 265o from A to B. What bearing would be taken from B to return to A?
A. 005 o
B. 085o
C. 090o
D. 355o
E. 275o
8. From a point on a cliff 200 m above sea level, the angle of depression to a boat is 40 o . The distance from the foot of the cliff to the boat to the nearest metre is:
A. 238 m
B. 168 m
C. 153 m
D. 261 m
E. 311 m
9. A boat sails from a harbour on a bearing of 045o for 100 km. It then takes a bearing of 190 o for 50 km. How far from the harbour is it, correct to the nearest km?
A. 51 km
B. 82 km
C. 66 km
D. 74 km
E. 3437 km
Draw appropriate labelled diagrams to solve Questions 6 –10
10. A hiker walks 3.2 km on a bearing of 1200 and then takes a bearing of 055 0 and walks 6 km. His bearing from the start is:
A. 013 o
B. 077 o
C. 235 o
D. 257 o
E. 330 o
11. A right pyramid with a square base is shown in the diagram. Each edge of the square base has length 8 cm and the height of the pyramid is 16 cm. The length of a sloping edge of the pyramid in centimetres is:
A. 288
B. 155
C. 125
D. 324
E. 425
12. In the figure ABCD the magnitude of angle BCA is closest to
A. 19°
B. 39°
C. 51°
D. 55°
E. 59°
13. A tree is growing near the block of land. The base of the tree, T, is at the same level as the corners, P and S, of the block of land. From point S, the angle of elevation to the top of the tree is 22°.
Calculate the height of the tree in metres, correct to one decimal place.
14. Chris leaves island A in a boat and sails 142 km on a bearing of 078° to island B. Chris then sails on a bearing of 191° for 220 km to island C, as shown in the diagram.
(a) Show that the distance from island C to island A is approximately 210 km.
Write your response on the Quiz Completion report (you need to download it online) and submit it as work submission for Week 4 [W04].
Submit it online, you can find the link
(b) Chris wants to sail from island C directly to island A. On what bearing should Chris sail? Give your answer correct to the nearest degree.
15. The base of a lighthouse D, is at the top of a cliff 168 metres above sea level. The angle of depression from D to a boat at C is 28o . The boat heads towards the base of the cliff, A, and stops at B. The distance AB is 126 metres.
(a) What is the angle of depression from D to B, correct to the nearest degree?
(b) How far did the boat travel from C to B, correct to the nearest metre?
Questions 1 to 9 and Fill in the blanks Questions 10 to 12
1. An arc of length 12 cm subtends an angle of magnitude θ° at the centre of a circle of radius length 7 cm. The value of θ, to the nearest degree, is:
A 98
B 127
C 2
D 118
E 84
2. Arc PQ of length 7 mm subtends an angle of 18° at the centre O of the circle.
The radius of the circle is:
A mm
35
π
B mm
70
π
C 35 mm
π
D 70 mm
π
E mm
18
π
3. Two places with the same longitude have latitude 26°N and 34°S. The distance between them is closest to
A 6000 km
B 6400 km
C 6700 km
D 5760 km
E 5420 km
4. The radius of the small circle which is the parallel of latitude 34°N is closest to
A 4906 km
B 4820 km
C 5305 km
D 5339 km
E 5430 km
5. The time difference between (45°S, 2°E) and (32°S, 65°E) is about
A 1 hour
B 2 hours
C 3 hours
D 4 hours
E 5 hours
6. The coordinates of a city are (53°N, 6°W). What are the coordinates of another location which is 16° east of this city?
A (53°N, 16°W)
B (69°N, 6°W)
C (53°N, 10°E)
D (53°N, 16°E)
E (37°N, 6°W)
7. The distance between the locations (34°N, 6°E) and (34°N, 36°E) passing along the parallel of latitude which passes through the two locations is closest to
A 1506 km
B 1730 km
C 1920 km
D 3010 km
E 2780 km
8. The area of the unshaded region is 2 cm2 .
Triangle BOC has a right angle at O.
The radius of the circle is closest to
A 0.9 cm
B 1.6 cm
C 2.2 cm
D 2.6 cm
E 3.6 cm
9. The solid OPQR, as shown opposite, is one-eighth of a sphere of radius 15 cm.
The point O is the centre of the sphere and the points P, Q and R are on the surface of the sphere.
∠POQ = ∠QOR = ∠ROP = 90°
The total surface area of the solid OPQR, in cm2, is closest to:
A 619
B 648
C 706
D 884
E 1767
10. Melbourne and Hokkaido Island in Japan lie on the same line of longitude and are 9094 km apart. If Melbourne has a latitude of 37. 83° S, find the latitude of Hokkaido Island, which is north of Melbourne. Give your answer correct to 2 decimal places.
Answer: _________° N
The following information relates to questions 11 and 12.
A cargo ship travels from Perth (115°E longitude and latitude 33°S) to Cape Town, South Africa (20° E longitude and latitude 33°S). The ship set sail at 4.00 am on 3 April and travelled at an average speed of 15 m/s.
11. If the journey from Perth to Cape Town was 8700 km, calculate the duration of the trip correct to the nearest minute.
Answer: __________ minutes
12a Find the date and time in Perth when the ship arrived at Cape Town.
b Find the time difference in hours between Perth and Cape Town.
c Find the date and local time of the ship’s arrival in Cape Town.
Answers:
12a. ______ April ______ ______(a.m. / p.m.)
12b. The time difference between Perth and Cape Town is ______ hours
12c. ______ April ______ ______(a.m. / p.m.)
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