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GEOMETRY, FUR1 2020 VCAA 7 MC

Cape Town has latitude 34° S and longitude 18° E.

Stockholm has latitude 59° N and longitude 18° E.

Assume that the radius of Earth is 6400 km.

The shortest distance along the meridian between Cape Town and Stockholm, in kilometres, is closest to

  1.   2011
  2.   3798
  3.   4021
  4.   6590
  5. 10 388
Show Answers Only

`E`

Show Worked Solution


 

`text(Distance)` `= {(59 + 34)}/360 xx 2pi xx 6400`
  `= 10\ 388.19…\ text(km)`

 
`=>  E`

GEOMETRY, FUR1-NHT 2019 VCAA 5 MC

The cities of Lima and Washington, DC have the same longitude of 77° W.

The shortest great circle distance between Lima and Washington, DC is 5697 km.

Assume that the radius of Earth is 6400 km.

Lima has a latitude of 12° S and is located due south of Washington, DC.

What is the latitude of Washington, DC?

  1. 39° N
  2. 51° S
  3. 51° N
  4. 63° N
  5. 65° S
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`A`

Show Worked Solution

     
 

`text(Arc Distance) \ WL` `= 5697`
`({theta + 12}/{360}) xx 2 xx pi xx 6400` `= 5697`
`theta + 12` `= (5697 xx 360)/(2 pi xx 6400)`
  `= 51°`
`:. \ theta` `= 39°`

 
`=> \ A`

GEOMETRY, FUR2 2019 VCAA 2

A cargo ship travels from Magadan (60° N, 151° E) to Sydney (34° S, 151° E).

  1. Explain, with reference to the information provided, how we know that Sydney is closer to the equator than Magadan.   (1 mark)
  2. Assume that the radius of Earth is 6400 km.

      

    Find the shortest great circle distance between Magadan and Sydney.

      

    Round your answer to the nearest kilometre.  (1 mark)

  3. The cargo ship left Sydney (34° S, 151° E) at 6 am on 1 June and arrived in Perth (32° S, 116° E) at 10 am on 11 June.

      

    There is a two-hour time difference between Sydney and Perth at that time of year.

      

    How many hours did it take the cargo ship to travel from Sydney to Perth?  (1 mark)

Show Answers Only
  1. `text(See Worked Solutions)`
  2. `10\ 500\ text(km)`
  3. `246\ text(hours)`
Show Worked Solution
a.  

`text{Sydney’s angle with the equator (34°S) is smaller than Magadan’s}`

`text{(60°N) resulting in a shorter distance to the equator.)`

 

b.   `text(Distance)` `= (34 + 60)/360 xx 2pi xx 6400`
    `~~ 10\ 500\ text(km)`

 

c.    `text(Perth is 2 hours behind Sydney)`

`text(6 am Sydney = 4 am Perth)`

`:.\ text(Travel time)` `= 4\ text{am (1 Jun)} – 10\ text{am (11 Jun)}`
  `= 6\ text(hours) + 10\ text(days)`
  `= 6 + 10 xx 24`
  `= 246\ text(hours)`

GEOMETRY, FUR1 2018 VCAA 3 MC

The city of Karachi in Pakistan has latitude 25° N and longitude 67° E.

Assume that the radius of Earth is 6400 km.

The shortest distance along the surface of Earth between Karachi and the North Pole, in kilometres, can be found by evaluating which one of the following products?

  1. `23/100 xx pi xx 6400`
  2. `25/180 xx pi xx 6400`
  3. `65/180 xx pi xx 6400`
  4. `67/180 xx pi xx 6400`
  5. `23/360 xx pi xx 6400`
Show Answers Only

`C`

Show Worked Solution

`text(Full circumference) = 2pi xx 6400`

♦ Mean mark 43%.

`:.\ text(Distance between Karachi and North Pole)`

`= 65/360 xx 2pi xx 6400`

`= 65/180 xx pi xx 6400`
 

`=> C`

GEOMETRY, FUR2 2017 VCAA 2

Miki will travel from Melbourne (38° S, 145° E) to Tokyo (36° N, 140° E) on Wednesday, 20 December.

The flight will leave Melbourne at 11.20 am, and will take 10 hours and 40 minutes to reach Tokyo.

The time difference between Melbourne and Tokyo is two hours at that time of year.

  1. On what day and at what time will Miki arrive in Tokyo?  (1 mark)

Miki will travel by train from Tokyo to Nemuro and she will stay in a hostel when she arrives.

The hostel is located 186 m north and 50 m west of the Nemuro railway station.

    1. What distance will Miki have to walk if she were to walk in a straight line from the Nemuro railway station to the hostel?

       

      Round your answer to the nearest metre.  (1 mark)

    2. What is the three-figure bearing of the hostel from the Nemuro railway station?

       

      Round your answer to the nearest degree.  (1 mark)


The city of Nemuro is located 43° N, 145° E.

Assume that the radius of Earth is 6400 km.

  1. The small circle of Earth at latitude 43° N is shown in the diagram below.
     
             
     
    What is the radius of the small circle of Earth at latitude 43° N?

     

    Round your answer to the nearest kilometre.  (1 mark)

  2. Find the shortest great circle distance between Melbourne (38° S, 145° E) and Nemuro (43° N, 145° E).

     

    Round your answer to the nearest kilometre.  (1 mark)

Show Answers Only
  1. `8\ text(pm (Wed))`
  2. i. `193\ text(m  (nearest metre))`
  3. ii. `345^@`
  4. `4681\ text(km  (nearest km))`
  5. `9048\ text(km  (nearest km))`
Show Worked Solution

a.   `text{Flight arrival (in Melb time) = 11:20 + 10:40 = 22:00 (Wed)}`

♦ Mean mark 40%.
COMMENT: A surprisingly poor result for this standard question.

`text(Tokyo time)` `=\ text(Melb time less 2 hrs)`
  `= 20:00\ (text(Wed))`
  `= 8\ text(pm (Wed))`

 

b.i.  `text(Using Pythagoras:)`

`d` `= sqrt(186^2 + 50^2)`
  `= 192.603…`
  `= 193\ text(m  (nearest metre))`

 

♦♦ Mean mark part (b)(ii) 27%.
MARKER’S COMMENT: N15°W is not a 3-figure bearing and received no marks.

b.ii.    `tan theta` `= 50/186`
  `theta` `= 15.04…^@`

 

`:. text(Bearing of)\ H\ text(from)\ N`

`= 360 – 15`

`= 345^@`

 

c.   `text(Let)\ \ x = text(radius of small circle)`

♦ Mean mark part (c) 43%

`sin47^@` `= x/6400`
`:.x` `= 6400 xx sin47^@`
  `= 4680.66…`
  `= 4681\ text(km  (nearest km))`

 

d.   

`text(Shortest distance)`

♦ Mean mark part (d) 38%

`= text(Arc length)\ NM`

`= 81/360 xx 2 xx pi xx 6400`

`= 9047.78…`

`= 9048\ text(km  (nearest km))`

GEOMETRY, FUR1 SM-Bank 32 MC

Makoua and Macapá are two towns on the equator.

The longitude of Makoua is 16°E and the longitude of Macapá is 52°W.

How far apart are these two towns if the radius of Earth is approximately 6400 km?

A.  `4000\ text(km)`

B.  `7600\ text(km)`

C.  `1\ 367\ 200\ text(km)`

D.  `1\ 447\ 600\ text(km)`

E.   `2\ 734\ 400\ text(km)`

Show Answers Only

`B`

Show Worked Solution

2UG-2006-19MC Answer

`text(Angular Difference)` `= 52 + 16`
  `= 68°`

 

`text(Arc)\ AB` `=\ text(Distance between two towns)`
  `= 68/360 xx 2 xx pi xx r`
  `= 68/360 xx 2 xx pi xx 6400`
  `= 7595.67…\ text(km)`

 
`=>  B`

GEOMETRY, FUR1 SM-Bank 15 MC

Which expression will give the shortest distance, in kilometres, between Mount Isa (20°S  140°E)  and Tokyo (35°N  140°E)?

  1. `15/360 xx 2 xx pi xx 6400`
  2. `55/360 xx 2 xx pi xx 6400`
  3. `125/360 xx 2 xx pi xx 6400`
  4. `140/360 xx 2 xx pi xx 6400`
  5. `305/360 xx 2 xx pi xx 6400`
Show Answers Only

`B`

Show Worked Solution

`text(Tokyo is 35° North of the equator, Mt Isa 20° South)`

`text(Arc length) = 55/360 xx 2 xx pi xx 6400`

`=>  B`

GEOMETRY, FUR2 SM-Bank 26

Two cities lie on the same meridian of longitude. One is 40° north of the other.

What is the distance between the two cities, correct to the nearest kilometre?  (2 marks)

Show Answers Only

`4468\ text{km   (nearest km)}`

Show Worked Solution

2UG 2015 23c Answer

`text(Distance between two cities)`

`=\ text(Arc length)\  AB`

`= 40/360 xx 2 xx pi xx r`

`= 1/9 xx 2 xx pi xx 6400`

`= 4468.04…`

`= 4468\ text{km  (nearest km)}`

GEOMETRY, FUR2 SM-Bank 14

Pontianak has a longitude of 109°E, and Jarvis Island has a longitude of 160°W.

Both places lie on the Equator

  1. Find the shortest great circle distance between these two places, to the nearest kilometre. You may assume that the radius of the Earth is 6400 km.    (2 marks)
  2. The position of Rabaul is 4° to the south and 48° to the west of Jarvis Island. What is the latitude and longitude of Rabaul?     (1 mark)
Show Answers Only
  1. `10\ 165\ text(km)\ \ \ text{(nearest km)}`
  2. `152^@ text(E)`
Show Worked Solution
a.   `text(Longitude difference)` `= 109 + 160`
    `= 269^@`

 

`=> text(Shortest distance)\ text{(by degree)}` `= 360 – 269`
  `= 91^@`

 

`:.\ text(Shortest distance)` `= 91/360 xx 2 pi r`
  `= 91/360 xx 2 xx pi xx 6400`
  `= 10\ 164.79…`
  `=10\ 165\ text(km)\ text{(nearest km)}`

 

b.   `text(Latitude)`
  `4^@\ text(South of Jarvis Island)`
  `text(S)text(ince Jarvis Island is on equator)`
  `=>\ text(Latitude is)\ 4^@ text(S)`
   
  `text(Longitude)`
  `text(Jarvis Island is)\ 160^@ text(W)`
  `text(Rubail is)\ 48^@\ text(West of Jarvis Island, or 208° West)`
  `text(which is)\ 28^@\ text{past meridian (180°)}`

 

`=>\ text(Longitude)` `= (180\ -28)^@ text(E)`
  `= 152^@ text(E)`

 

`:.\ text(Position is)\ (4^@text{S}, 152^@text{E})`

GEOMETRY, FUR2 SM-Bank 12

This diagram represents Earth. `O` is at the centre, and `A` and `B` are points on the surface.
 

2UG-2005-27b
 

Find the shortest great circle distance from `A` to `B`.

Give your answer in to the nearest km.   (2 marks)

Show Answers Only

`text{4803 km  (nearest km)}`

Show Worked Solution

`A : 35^@text(N)\ 20^@text(E)\ \ \ \ B:8^@text(S)\ 20^@text(E)`

HSC 2005 27b

`text(Angular difference)` `= 35^@ + 8^@`
  `= 43^@`

 

`:.\ text(Distance from)\ A\ text(to)\ B`

`= 43/360 xx 2 xx pi xx r`

`= 43/360 xx 2 xx pi xx 6400`

`= 4803.1…`

`= 4803\ text{km  (nearest km)}`

GEOMETRY, FUR2 SM-Bank 11

An aircraft travels at an average speed of  913 km/h. It departs from a town in Kenya  (0°, 38°E)  on Tuesday at 10 pm and flies east to a town in Borneo  (0°, 113°E).

  1. What is the distance, to the nearest kilometre, between the two towns? (Assume the radius of Earth is  6400  km.)  (2 marks)
  2. How long will the flight take? (Answer to the nearest hour.)   (1 mark)
  3. What will be the local time in Borneo when the aircraft arrives? (Ignore time zones.)   (1 marks)
Show Answers Only
  1. `8378\ text(km)`
  2. `9\ text(hours)`
  3. `text(12 midday on Wednesday)`
Show Worked Solution
a.   `text(Angular difference in longitude)`

`= 113 – 38`

`= 75^@`
 

`text(Arc length)` `= 75/360 xx 2 xx pi xx 6400`
  `= 8377.58…`
  `= 8378\ text(km)\ text{(nearest km)}`

 
`:.\ text(The distance between the two towns is 8378 km.)`

 

b.    `text(Flight time)` `= text(Distance)/text(Speed)`
    `= 8378/913`
    `= 9.176…`
    `= 9\ text(hours)\ text{(nearest hr)}`

 

c.   `text(Time Difference)` `= 75 xx 4`
    `= 300\ text(minutes)`
    `= 5\ text(hours)`

  
`text(S)text(ince Kenya is further East,)`

`=>\ text(Kenya is +5 hours)`

 

`:.\ text(Arrival time in Kenya)`

`= text{10 pm (Tues) + 5 hrs + 9 hrs}\ text{(flight)}`

`= 12\ text(midday on Wednesday)`

GEOMETRY, FUR2 2016 VCAA 3

A golf tournament is played in St Andrews, Scotland, at location 56° N, 3° W.

  1. Assume that the radius of Earth is 6400 km.

     

    Find the shortest great circle distance to the equator from St Andrews.

     

    Round your answer to the nearest kilometre.  (1 mark)

  2. The tournament begins on Thursday at 6.32 am in St Andrews, Scotland.

     

    Many people in Melbourne will watch the tournament live on television.

     

    Assume that the time difference between Melbourne (38° S, 145° E) and St Andrews (56° N, 3° W) is 10 hours.

     

    On what day and at what time will the tournament begin in Melbourne?  (1 mark) 

Show Answers Only
  1. `6255\ text{km (nearest km)}`
  2. `4:32\ text(pm on Thursday)`
Show Worked Solution
a.   

`text(Great circle distance to equator)`

♦ Mean mark 42%.
MARKER’S COMMENT: Longitude figures were not relevant here!

`= theta/360 xx 2pir`

`= 56/360 xx 2 xx pi xx 6400`

`= 6255.26…`

`= 6255\ text{km (nearest km)}`

 

b.   `text(Melbourne is East of St Andrews)`

♦♦ Mean mark 37%.
MARKER’S COMMENT: The time difference between the two cities here is given! Longitudinal difference calculations are not required.

`=>\ text(Melbourne is 10 hours ahead.)`

`:.\ text(Time in Melbourne)`

`= 6:32\ text(am + 10 hours)`

`= 4:32\ text(pm on Thursday)`