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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, 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))`