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GEOMETRY, FUR1 2020 VCAA 6 MC

The cities of Lhasa (30° N, 91° E), Chengdu (31° N, 104° E) and Shanghai (31° N, 121° E) are all in the same time zone.

Assume that 15° of longitude equates to a one-hour time difference.

Which one of the following statements is true?

  1. The sun rises at the same time in all three cities.
  2. The sun rises in Lhasa approximately one hour before it rises in Chengdu.
  3. The sun rises in Lhasa approximately two hours after it rises in Shanghai.
  4. Sunrise in each of the three cities is approximately 15 minutes apart.
  5. The sun rises in Shanghai and Chengdu at the same time but it rises earlier in Lhasa.
Show Answers Only

`C`

Show Worked Solution

`text(Consider longitudinal differences:)`

`text(Lhasa to Chengdu) = 104 – 91 = 13^@`

`→ text(Time difference) = 13 xx 4 = 42\ text(mins)`

`text(Lhasa to Shanghai) = 121 – 91 = 30^@`

`→ text(Time difference) = 30 xx 4 = 120\ text(mins)`

`:.\ text(Sun rises in Lhasa approx 2 hours after Shanghai)`

`=>  C`

GEOMETRY, FUR1 2020 VCAA 1 MC

Each year begins on 1 January.

The location that begins each year first is

  1. Kathmandu (28° N, 85° E).
  2. Cairo (30° N, 31° E).
  3. Kabul (34° N, 69° E).
  4. Nairobi (1° S, 37° E).
  5. Port Moresby (9° S, 147° E).
Show Answers Only

`E`

Show Worked Solution

`text(Location that is first has the highest “East”)`

`text{longitude (can’t be greater than 180°).}`

`=>  E`

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 2019 VCAA 6 MC

All cities in China are in the same time zone.

Wuhan (31° N, 114° E) and Chengdu (31° N, 104° E) are both cities in China.

On one day in January, the sun set in Wuhan at 5.50 pm.

Assuming that 15° of longitude equates to a one-hour time difference, the time the sun set in Chengdu on the same day is

  1. 4.20 pm
  2. 5.10 pm
  3. 5.50 pm
  4. 6.30 pm
  5. 7.20 pm
Show Answers Only

`D`

Show Worked Solution

`text(Longitudinal difference)`

`= 114 – 104`

`= 10^@`
 

`text(Time difference)` `= 10/15 xx 60`
  `= 40\ text(minutes)`

 
`text(Wuhan is further East)`

`=>\ text(Chengdu 40 minutes behind)`
  

`:.\ text(Sunset in Chengdu)`

`= 5:50\ text(pm) + 40\ text(mins)`

`= 6:30\ text(pm)`
 

`=>  D`

GEOMETRY, FUR2 2018 VCAA 2

Frank travelled from Melbourne (38° S, 145° E) to a tennis tournament in Ho Chi Minh City, Vietnam, (11° N, 107° E).

Frank departed Melbourne at 10.30 pm on Monday, 5 February 2018.

Frank arrived in Ho Chi Minh City at 8.00 am on Tuesday, 6 February 2019.

The time difference between Melbourne and Ho Chi Minh City is four hours.

  1.  How long did it take Frank to travel from Melbourne to Ho Chi Minh City?

     

     Give your answer in hours and minutes.   (1 mark)

Ho Chi Minh City is located at latitude 11° N and longitude 107° E.

Assume that the radius of Earth is 6400 km.

  1. i. Write a calculation that shows that the radius of the small circle of Earth at latitude 11° N is 6282 km, rounded to the nearest kilometre.  (1 mark)
  2. ii. Iloilo City, in the Philippines, is located at latitude 11° N and longitude 123° E.

     

        Find the shortest small circle distance between Ho Chi Minh City and Iloilo City.

     

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

Show Answers Only
    1. `text(13 hours 30 min.)`
    2. i. `text(See Worked Solutions)`
    3. ii. `1754\ text(km)`
Show Worked Solution

a.   `text{8am Ho Chi Minh (Tues) = 12 midday Melbourne (Tues)}`

♦♦ Mean mark 29%.
COMMENT: Review carefully!

`:.\ text(Travel time)` `=\ text{10:30 pm (Mon) to 12 midday (Tues)}`
  `=\ text(13 hours 30 min.)`

 

b.i.  `text(Let)\ \ x=\ text(small circle radius)`

Mean mark part (b)(i) 52%.

 


 

`cos 11^@` `= x/6400`
`:. x` `= 6400 xx cos 11^@`

 

b.ii.  `text(Iloilo City)\ ->\ text(same latitude)`

♦♦ Mean mark part (b)(ii) 33%.
MARKER’S COMMENT: Many students incorrectly used 6400 as the radius in part (b)(ii). Understand why  `6400 xx cos 11°`  is the correct radius here!

 `text(Longitudinal difference) = 123 – 107 = 16^@`
 


 

`:.\ text{Small circle distance (arc)}`

`= 16/360 xx 2 xx pi xx (6400 xx cos 11^@)`

`= 1754.38…`

`= 1754\ text{km (nearest km)}`

GEOMETRY, FUR1 2018 VCAA 5 MC

On the first day of June 2019 in the city of St Petersburg, Russia, (60° N, 30° E) the sun is expected to rise at 3.48 am.

On this same day, the sun is expected to rise in Helsinki, Finland, (60° N, 25° E) approximately

  1. at the same time.
  2. 20 minutes earlier.
  3. 20 minutes later.
  4. one hour earlier.
  5. one hour later.
Show Answers Only

`C`

Show Worked Solution

`text(Difference in longitude) = 30 – 25 = 5°`

Mean mark 51%.

`text(S)text(ince 15° = 1 hour difference)`

`text(Time difference)` `= 5/15 xx 60`
  `= 20\ text(minutes)`

  
`text(St Petersburg is further east.)`

`:.\ text(Sunrise in Helsinki is 20 mins later.)`

`=> 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 2017 VCAA 3 MC

The locations of four cities are given below.

Adelaide (35° S, 139° E)     Buenos Aires (35° S, 58° W) 
Melilla (35° N, 3° W)      Heraklion (35° N, 25° E)

 

In which order, from first to last, will the sun rise in these cities on New Year’s Day 2018?

  1. Adelaide, Heraklion, Buenos Aires, Melilla
  2. Adelaide, Heraklion, Melilla, Buenos Aires
  3. Heraklion, Adelaide, Melilla, Buenos Aires
  4. Melilla, Adelaide, Heraklion, Buenos Aires
  5. Melilla, Adelaide, Buenos Aires, Heraklion
Show Answers Only

`B`

Show Worked Solution

`text(Sun rises in the East.)`

`text(Adelaide is further east than Heraklion.)`

`text(Melilla is further east (i.e. less west))`

`text(than Buenos Aires.)`

`:.\ text{Order is: Adelaide (139°E) , Heraklion (25°E),}`

    `text{Melilla (3°W), Buenos Aires (58°W)}`

`=> B`

GEOMETRY, FUR1 SM-Bank 35 MC

Kim lives in Perth (32°S, 115°E). He wants to watch an ice hockey game being played in Toronto  (44°N, 80°W)  starting at 10.00 pm on Wednesday.

What is the time in Perth when the game starts?

A.   9.00 am on Wednesday

B.   7.40 pm on Wednesday 

C.   9.00 pm on Wednesday

D.   12.20 am on Thursday

E.   11.00 am on Thursday

Show Answers Only

`E`

Show Worked Solution

`text(Perth is East of Toronto) =>\ text(Ahead)`

`text(Longitudinal difference)`

`= 115 + 80`

`= 195^@`
 

`text(Time difference)` `= 195/15`
  `= 13\ text(hours)`

 

`:.\ text(Time in Perth)`

`=\ text{10 pm (Wed) + 13 hours}`

`=\ text(11 am on Thursday)`
 

`=>  E`

GEOMETRY, FUR1 SM-Bank 18 MC

Stockholm is located at  59°N 18°E  and Darwin is located at  13°S 131°E. 

What is the time difference between Stockholm and Darwin? (Ignore time zones and daylight saving.)

  1. 184 minutes
  2. 188 minutes
  3. 288 minutes
  4. 452 minutes
  5. 596 minutes
Show Answers Only

`D`

Show Worked Solution

`text(Stockholm is 59°N 18°E,  Darwin is 13°S 131°E)`

`text{Angular difference (longitude)}`

`= 131^@− 18^@`

`= 113^@`
 

`:.\ text(Time difference)` `= 113 xx 4`
  `= 452\ text(minutes)`

 
`⇒ D`

GEOMETRY, FUR2 SM-Bank 15

Osaka is at  34°N, 135°E, and Denver is at  40°N, 105°W. 

  1. Show that there is a 16-hour time difference between the two cities.
    (Ignore time zones.)    (1 mark)
  2. John lives in Denver and wants to ring a friend in Osaka. In Denver it is 9 pm Monday.     

     

    What time and day is it in Osaka then?   (1 mark)

  3. John’s friend in Osaka sent him a text message which happened to take 14 hours to reach him. It was sent at 10 am Thursday, Osaka time.

     

    What was the time and day in Denver when John received the text?    (1 mark)

Show Answers Only
  1. `text{Proof (See Worked Solutions)}`
  2. `1\ text(pm Tuesday)`
  3. `8\ text(am Thursday)`
Show Worked Solution
a.   `text(Longitude difference)` `= 135 + 105`
    `= 240^@`

 

`text(Time difference)` `= 240/15`
  `= 16\ text(hours   … as required)`

 

b.  `text(Denver is behind Osaka time because it is further west.)`

`:.\ text(Time in Osaka)` `= 9\ text(pm Monday plus 16 hours)`
  `= 1\ text(pm Tuesday)`

 

c.  `text(Denver is 16 hours behind Osaka)`

`:.\ text(John will receive the text at 10 am Thursday less 16)`

`text{plus 14 hours (i.e. 8 am Thursday.)}`

GEOMETRY, FUR2 SM-Bank 13

Melbourne is located at (38°S, 145°E) and Dubai is located at (24°N, 55°E).

  1. Calculate the difference in longitude between Melbourne and Dubai.  (1 mark)
  2. Show that the time difference between Melbourne and Dubai is 6 hours.  (1 mark)
  3. A plane leaves Melbourne on Friday at 11.30 pm. The flight time to Dubai is 15 hours.

     

    What will be the time and the day in Dubai when the plane is due to land?  (1 marks)

Show Answers Only
  1. `90^@`
  2. `text(See Worked Solutions)`
  3. `text(8:30 am on Saturday)`
Show Worked Solution

a.   `text(Difference in longitude)`

`= 145 – 55`

`= 90^@`

 

b.   `text(S)text(ince)\ 1^@ = 4\ text(min time difference,)`

`text(Time difference)`

`= 90 xx 4`

`= 360\ text(mins)`

`= 6\ text(hours  … as required)`

 

c.    `text(Arrival time)` `= 11:30 + 15\ text(hours)`
    `= 14:30\ text{(Melbourne time)}`

 

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

`= 14:30 – 6\ text(hours)`

`= 8:30\ text(am on Saturday)`

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)`

GEOMETRY, FUR1 2016 VCAA 4 MC

All towns in the state of Victoria are in the same time zone.

Mallacoota (38°S, 150°E) and Portland (38°S, 142°E) are two coastal towns in the state of Victoria.

On one day in January, the sun rose in Mallacoota at 6.03 am.

Assuming that 15° of longitude equates to a one-hour time difference, the time that the sun was expected to rise in Portland is

  1. 5.31 am.
  2. 5.55 am.
  3. 6.03 am.
  4. 6.11 am.
  5. 6.35 am.
Show Answers Only

`E`

Show Worked Solution
`text(Angular difference)` `= 150 – 142`
  `= 8^@`

 

`text(Time difference)` `= 8/15 xx 60`
  `= 32\ text(minutes)`
♦♦♦ Mean mark 26%.

 

`text(S)text(ince Portland is West of Mallacoota,)`

`text(it’s sunrise will be later.)`
 

`:.\ text(Portland)` `= 6:03 + 32\ text(minutes)`
  `= 6:35\ text(am)`

 
`=> E`