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.

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 easy one 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, 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

*Of course you might argue that the tourists will still not be harmed as long as the gradient ≠ −√3 or but let's just take it that there're more people and boats beyond −√3 or shall we?

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 puke water spout whenever you see a question with a y²=kx curve!

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。。。