Unfortunately, Miss Loi thinks your Ron Ron has run away with mathslover <333 already (since she's the one who helped him find his wrong bearing).

HAHA What a way to ensure that students desperately solve the question. I should totally solve this now.

Or not! I shall not solve it for fear of him taking you if he catches up with you! I shall let him catch me instead. Yup!

Bear in mind that in maps & scales questions that typically involve large multiples e.g. 1 : 50 000, 1 : 250 000 etc. (which will get even larger when it comes to areas), it's always safer in terms of accuracy to convert first.

Say we wish to calculate the actual area of ΔXYZ (to the nearest whole no.) in our question:

• If we convert all map values to metres from the start,
  Area of ΔXYZ = (0.5)(300)(250)sin120° = 32475.95 m² ≈ 32476 m²

• But if we use the original map values in our calculation and then convert the final answer via the scale:
  1 cm → 50 m
  (1 cm)² → (50 m)²
  - note the brackets acting as an anti-careless mistake measure
  1 cm² → 2500 m²

  Map Area of ΔXYZ = (0.5)(6)(5)sin120° = 12.99038106 ... cm²

  But if we somehow cleared our calculator and use just 4 sig. fig. (which some might do since we only need at least 4 sig. fig. for intermediate workings) of this answer, the actual area becomes 12.99 × 2500 = 32475 m² which is 1 m² short of the actual answer!

Miss Loi used the map values for the longish Cosine Rule calculation in part i. simply to avoid having to key in big numbers in the calculator (esp. for those who have the habit of 'breaking up' their calculations in their calculator), but to be safe she used the full value of 9.539392014 stored in her calculator (instead of truncating it to lesser sig. fig.) for the conversion (see updated working above).

So it's an issue of speed vs accuracy and it depends on the scenario and how comfortable one is with the subject (and calculator skills).

Thanks once again for bringing this up

He flung his GPS straight at the confused henchman who still had no idea he caught the wrong woman.

"Not this one! That sexy one who was running barefoot! Can't you tell when someone is trying to escape?"

But there is still hope! The Ah Beng silently notes down the address of Miss Loi's relative. His multiple henchmen will lay in watch for Miss Loi's next visit... ...

炸雷惊响，刚还是万里晴空，不知何时，竟下起了滂沱大雨。

___________________________________________

Miss Loi 借题发挥 suan 我! Next time must double check before posting

@ Miss Loi:

But, if the question is regarding area (say if we need to find actual area of XYZ in the question) or volume would you suggest converting first? Since some students are likely to forget that conversion from cm² to m² is divide by 10000, not 100.

Though yes, it is less tedious to use the smaller map values for 1 dimensional measures like that in the question.

Recalling a past lesson by the Maths Tutor, the Ah Beng recognized that this Three-Dimensional Trigonometry problem (coupled with bearings, speed and scales & maps) required him to model the problem and draw a clearly-labelled diagram in order to help him visualize the situation.

Unfortunately he didn't have a pen and paper with him so this was the 3D image he formed in his head:

3D View (Pics of Miss Loi, the henchman and the car are optional)

To obtain the actual distance of YZ part (i), he saw from the diagram below that ∠YXZ = 210° − 90° = 120° (since Y is due East of X).

View from Top

And thus applying the Cosine Rule to ΔXYZ like what mathslover has done, he obtained

YZ² = XY² + XZ² − 2(XZ)(XY)cos 120°
= 5² + 6² - 2(5)(6)cos 120°
= 91
⇒ YZ = √91 = 9.539392014 cm

And since the scale is 1 : 5000
⇒ 1 cm → 5000 cm
⇒ 1 cm → 50 m
⇒ actual distance of YZ = 9.539392014 × 50 m ≈ 477 m (3 s.f.)

Note: It's arguably simpler in this case to perform the initial calculation using the map values of 5 cm and 6 cm, as we only need to perform the conversion to metres once for the final answer. This way, there are less digits to key into your calculator too

To determine who reached the car at Z first (after he had obtained the distance of YZ) he looked back to his initial 3D diagram above and found that there was still one piece of missing information i.e. the distance TX that the lift traveled in the Maths Tutor's two-legged journey.

Side View

Using the given angle of elevation (i.e. looking up) from Y to to T, he sketched the 'side view' diagram (see above) in his head (with help from mathslover's subliminal advice) and easily obtained

TX = (5 × 50 m) tan 17° = 76.43 m

So to recount mathslover's workings:

• Elevator’s speed = 1.9 m/s
  Elevator’s distance travelled (TX) = 76.43 m
  Elevator’s time taken along TX= 76.43 / 1.9 = 40.23 seconds

  Maths Tutor's speed = 2.5 m/s
  Maths Tutor's distance travelled (XZ) = 300 m
  Maths Tutor's time taken along XZ= 300 / 2.5 = 120 seconds

  ∴ Total time taken by Maths Tutor to reach Z = 120 + 40.23 = 160.23 seconds

• Henchman’s speed = 3 m/s
  Henchman’s distance (YZ) = 477.0 m (always use at least 1 s.f. more in intermediate workings for non-exact values else may get penalized!)
  Henchman’s time = 477.0 / 3 = 159 seconds

So in theory, the henchman reached the car 1.23 seconds earlier than the Maths Tutor

BUT ...

We haven't mentioned about the bearing in part ii. yet!

While the Ah Beng rightly

• chose to calculate the bearing of Z from Y since he needed to tell his henchman how to get to Z from Y
• knew that this bearing is measured at Y, clockwise from the North (see red text in diagram below)

• used the Sine Rule on ΔXYZ to obtain ∠XYZ as follows:

  
  ⇒ ∠XYZ = 33.0°

View from Top

he (or his subliminal advisor mathslover) perhaps neglected to look at the diagram hard enough and calculated the bearing of Z from Y as 180° + 33.0° = 213.0° where it should have been 180° + (90° − 33.0°) = 237.0°!!!

*checks diagram again and again just to be sure*

*evil laughter*

Using cosine rule,
YZ² = XY² + XZ² - 2(XZ)(XY)cos120
= 250² + 300² - 2(250)(300)cos120
= 227500
YZ = 477m (3 s.f.)

Using sine rule,
sin (XYZ) / 300 = sin (120) / 476.97
XYZ = 33.0 degrees
Bearing from Y = 180 + 33.0 = 213.0 degrees (1 d.p.)

Henchman's speed = 3m/s
Henchman's distance = 477m
Henchman's time = 477 / 3 = 159 seconds

Elevator's speed = 1.9m/s
Elevator's distance = 250m x tan 17 degrees = 76.43m
Elevator's time = 76.43 / 1.9 = 40.23 seconds

Miss Loi's speed = 2.5m/s
Miss Loi's distance = 300m
Miss Loi's time = 300 / 2.5 = 120 seconds

Total time taken by Miss Loi = 120 + 40.23 = 160.23 seconds

The henchman managed to reach the car 1.23 seconds before Miss Loi did. He watched as she rushed towards the car, panting, straddled with new year goodies, and waving her Jimmy Choo heels frantically.

He grinned.

Seeing the unfriendly-looking suspicious man loitering about her car, Miss Loi decided not to finish her last 3.075m of dash and instead turned 180 degrees and rushed back!

Unfortunately, as Miss Loi could only run at 2.5m/s but the henchman could run at 3m/s, he eventually caught up with her...