Continuing from last week’s episode …
The shadows lengthened rapidly as dusk began to fall.
The *clack clack clack* of the Maths Tutor’s killer 5-inch Jimmy Choo stilettos echoed off the concrete walls as she made her way towards the elevator down the corridor, each step an excruciating twist of an imaginary dagger deeper into his wounded heart.
The Ah Beng cut a lonely figure as he watched, forlornly, her lithe figure receding gradually into the distance … along with his hopes … leaving him with a profound feeling of loss …
But loss often leads to suffering, suffering leads to anger, and anger leads to hate. And suddenly a voice whispered in his head,
Oh my mighty Smiling 哥 of this sprawling GRC, how can you stomach this? No one here has ever said NO to you! What would the others say?
He eyes began to narrow as he unwittingly gripped the mandarin orange tighter and tighter …
Remember what the Maths Tutor taught her students? When all seem lost and no matter how difficult the exam is, never hand in a blank paper! At least write something to score some working marks! NEVER GIVE UP!!!
Consumed by the Dark Side rage, he crushed the poor mandarin orange and threw it violently to the floor (to the shock of the watching kaypoh aunties), and decided that not only will he NOT give up, he’ll make sure he gets his Maths Tutor by any means necessary.
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Meanwhile the sickening sound of that splattered mandarin orange startled the Maths Tutor, who couldn’t wait for the elevator door to open while the Ah Beng calmly began calling his henchmen on his phone …
As all his usual henchmen are busily gambling away at the new Sentosa Casino, the only one available is a young fresh ‘recruit’ who happens to be hanging around nearby.
From the GPS application on his phone, the Ah Beng marks out his HDB flat’s location as X. His lone henchman is currently situated at Y which is due east of X. He spots the Maths Tutor’s car parked at Z which is on a bearing 210° from X. X, Y and Z all lie on horizontal ground. The Maths Tutor enters the elevator at a point T which is directly and vertically above X, with the angle of elevation of T from Y being 17°. On his GPS map with a scale of 1 : 5000, XY = 5 cm and XZ = 6 cm.
To catch the Maths Tutor, the henchman needs to reach her car at Z before she does. But the henchman is unfamiliar with the territory, and only has a small 20-cent compass he extorted from a kid in the neighbourhood. Find
- the actual distance between Y and Z (in m).
- the bearing he must tell his henchman in order for the latter to locate the car.
Assuming the henchman sets off for the car at a constant speed of 3 m/s along YZ the moment the elevator (with the tutor on board) begins its uninterrupted descent from T at a constant speed of 1.9 m/s to the ground floor at X, afterwhich she removes her Jimmy Choos and runs straight for the car along XZ at a constant speed of 2.5 m/s, who will reach the car first?
Will Loi Loi escape? Or will Smiling 哥 finally have her in his clutches?
Stay tuned for the next episode after someone solves the (cheong hei) math questions above!
11 Comments
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Hmmm... more machinimas of Ah Beng and Miss Loi... hmmmmm!!
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FoxTwo, the Maths Tutor still hasn't recovered from the haunting memory of this!
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XY = 5000 x 5 cm = 25000 cm = 250 m
XZ = 5000 x 6 cm = 30000 cm = 300 m
Angle YXZ = 210 - 90 = 120 degrees.
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...
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O:
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Unfortunately, Vimeo deleted that video, and I didn't do a backup. So, we'll just have to remake the whole thing ALL OVER AGAIN!!!
*evil laughter*
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The question was long and cheong hei. The scenario was complex. Information was strewn in a mess throughout the question, like litter after the Chingay Parade.
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:
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).
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.
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
used the Sine Rule on ΔXYZ to obtain ∠XYZ as follows:
⇒ ∠XYZ = 33.0°
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*
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And so the Ah Beng watched in dismay as his young henchman ran all 477 m to the wrong car, grinned and ended up catching the wrong woman (coz there were others who were 拜年-ing with their new year goodies at the time too), while the Sexy Maths Tutor promptly jumped into her car to complete her great escape ... all because he neglected to look at his diagram properly and gave the wrong bearing i.e. direction as a result!
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“骗人！”The Ah Beng 怒发冲冠，凭走廊处。举起拳头，仰天长啸，满怀怨恨：“不可能！”
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。
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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. 🙂
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That's a good question brought up by mathslover.
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 😉
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And I am wondering why no one has commented on Ron Ng (<333!!!) yet.
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!
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@Jac: Wow a 知音 is here!
Unfortunately, Miss Loi thinks your Ron Ron has run away with mathslover <333 already (since she's the one who helped him find his
wrongbearing).