He was there when it all began.
Guarding fiercely the old mouth of the Singapore River, he witnessed the meteoric rise and rise of a glorious nation, as modern skyscrapers rose around him, and old warehouses turned into trendy dance clubs, and students began using calculators in their PSLE, while he dutifully maintained law and order in the river by keeping errant sampans in line with his lethal water spout.
As the years went by, however, he slowly felt forgotten by the relentless pace of progress, as the sampans themselves got transformed into tour boats.
He spouted a watery sigh of sadness while he looked on as one by one, his old friends like the Satay Club and the National Library went the way of the bulldozers.
More spouts of resignation were to follow when he learnt that younger and mightier clones (including a Godzilla-like monster that can shoot laser beams from its eyes) were recruited across the island to perform his job, and that the only occasions he was ever mentioned were in jokes involving drunk patrons of pubs and clubs.
But still he kept his dignity in defiance of the changing times, and continued his balancing act in a forgotten corner of Singapore to spray his jet of water with pride, even when they cruelly deprived him of his daily sunrise viewing by building a huge bridge across his face.
In the end, the straw that broke the Merlion’s back came when he was relocated against his will to another location. The loss of his spiritual home, combined with his inability to get used to being photographed and molested left, right and center by daily hordes of kawaii teeny trigger-happy tourists in this ironically spanking new location, finally pushed him over the edge.
And so, on a dark sunless day, he turned (as far as he could) to take a final look at his son (and discovered to his further horror that he hasn’t really grown all these years), before closing his eyes and imploring the heavens to put him out of his misery.
A passing 雷公 aka Lightning God heard his plea, and proceeded to plot his target coordinates …
The Merlion’s water spout follows a path defined by the curve y2 = 4x + 64, y > 0 and the lightning bolt (that travels in a straight line) is poised to strike the Merlion’s head at the point (0, 8), which lies on the path of the water spout.
On the seaward side, a group of tourists are snapping photos of the Merlion’s front from a Singapore River Cruise boat, with a line of vision that’s tangent to the water sprout at (0, 8).
On the landward side, a group of kawaii teeny Japanese tourists are snapping photos of the Merlion’s rear with an angle of elevation of 60° from them to the point (0, 8).
Given that the straight line lightning bolt is so powerful that it can travel through the Merlion’s head and harm anything beyond, find the range of possible values of its gradient so that the tourists will be safe from harm.
While this should ultimately be a straightforward enough problem involving the special y2 curve that made its way to the new A-Maths Syllabus last year, please make sure you sketch out the right graphs in this scenario in order to prevent the 雷公 from making any careless mistake that can lead to tragic consequences for the innocent tourists!
3 Comments
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find tangent to the curve at (0,8).
y2 = 4x+64
.
When x = 0,
.
equation of tangent to the curve:
when y = 0, x = -32
location of tourists on boat: (-32,0)
angle of elevation from tourists on boat to (0,8) =
= 14.04° (3sf)
Considering the triangle whose vertices are at (0,8), (-32,0) and the Japanese tourists, the angle which lightning can strike without touching them would be 180 - 60 - 14.04 = 105.96 degrees.
Graph: http://img9.imageshack.us/img9/2872/merlionf.jpg
P.S. Pardon me for humiliating our dear merlion further in the graph.
P.P.S. And hope Miss Loi doesn't mind a 不速之客 snatching her questions from her students.
看没有人回答，觉得鱼尾狮很可怜。。。
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Oh thank you so much, on behalf of the poor Merlion & 雷公, to you mathslover for helping put him out of his misery!
Actually this is a very simple and straightforward problem meant to highlight, apart from the usual angles of elevation, the curve of the form y2 = kx with examples as shown below:
It's worth noting from the diagram above that the direction of a y2 = kx curve will depend on whether the coefficient k of x is positive or negative.
So in the deliberate absence of a diagram
else the question would be too easyone must first determine the direction that the Merlion is facing.Since the coefficient of x in y2 = 4x + 64 is 4 which is POSITIVE, we know that the curve cum waterspout & Merlion must be facing left. And since it's given that the point (0, 8) lies on the curve (i.e. the water spout), we can deduce that the curve is shifted leftwards with the y-intercept at (0, 8).
Borrowing your cute Reggae Merlion with the rasta hairdo for a quick sketch, the situation becomes so much clearer ...
On the seaward side, the max gradient shall be the tangent of the waterspout (the y2 = 4y + 64 curve) at (0, 8), as you've correctly pointed out i.e.
(see working in Comment #1 above)
On the landward side, since we're already given that the angle of elevation is 60°, (see this diagram for reference) we can straightaway calculate the gradient via a simple ... hold your breath ...
tan 60°
, but sadly many students fail to realize this in the heat of the exam and often end up wasting precious time to calculate intersection points, coordinates etc.
Now tan 60° = √3 BUT make sure you insert a negative (-) sign in your final answer i.e. max gradient = −√3 since we've already determined from our quickly sketched diagram that the landward side is on the right so the gradient has to be negative.
∴ In order not to harm the tourists the gradient of the 雷公's lightning bolt has to be between −√3 and
😉
P.S. You probably realize by now that the question only asked for the gradient? Why so many coordinates and angles in your answer???
P.P.S. Oh don't worry about 'snatching' the question. Miss Loi thinks that many are simply waiting at the sidelines for the answer 🙁
Lastly, hope all students will remember the poor Merlion's
pukewater spout whenever you see a question with a y2=kx curve!曜
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Think I need a visit to the temple too 🙁
So careless! I kept thinking the question asked range of angles of the lightning.
As for the too-many coords.. I admit the graph was semi-plotted by joining significant coords (0,y) (x,0) to make a parabola-like-line (quadratic) because I basically totally forgot how it should look like 🙁
Thank you Miss Loi for oiling my rusty brain a little bit.