Hi, I am currently trying to do my PSMT for unit 3 and I have hit a roadblock. So basically we have to develop 3 models for a swimming race to show how the race plays out. We were given one function to work with which is:

d=0.005t^2-0.027t

They tell us that the race is 50 m and that the swimmers must finish within 15 seconds of each other.

From this, I have worked out the times and everything but I am stuck with the models.

We have to use a trigonometric function for one, a log function for another one, and an exponential for the last one to represent the swimmer's race.

The trigonometric one is where I'm stuck.

If I have two points that I need the graph to go through, how do I work out what the amplitude, period, and anything else is?

My points are (0,0) and (89,50), and i know it has to be a positive sine graph

Any help would be appreciated!

So, there's two ways to approach this question - the usual way, and the big brain way. The usual way is you know that the general equation for a sin(x) graph is \(f(x)=a\sin(b(x-h))+k\), you sub in the unknowns, and then solve simultaneously. Since you have four unknowns, you need four points - you have two, that's not going to work.

The big brain way? Think about what each of those parameters mean. You mentioned the amplitude and period, but I want to take about the translation parameters - h and k. Changing these will move the graph of f(x)=sin(x) around - but do we want to move this graph around? Think about it - the graph of f(x)=sin(x) ALREADY has the point (0,0) - so why bother changing the graph if it already works? So, if we set h=k=0, you now only have two parameters you need to find!

Cool, so amplitude and period - how the hell do we work with these? Well, if you sub our first point into the equation, we get:

\[

f(0)=0\\

a\sin(b\times 0)=0\\

a\times 0=0\\

0=0

\]

Ah, so turns out (0,0) is now a useless point to us. Well, let's think about (89,50). Technically, any part of the sin(x) curve could go through that point, so a and b could be ANYTHING... But what if we make (89,50) a maximum? If you think about it, you only want to go through the first quadrant of the sin(x) curve - because otherwise, that means the swimmer is going to move backwards, and that's not going to help you win a race. We could make it so any part of the curve in the first quadrant hits the point (89,50), too - but if you know that (89,50) is a maximum, you should be able to figure out incredibly quickly what the amplitude is. Then, you just need to change the frequency so that the first maximum happens when x=89.

Hopefully this makes sense - I know the last bit would make more sense if I showed you the equations, but I don't want to do EVERYTHING for you. Let me know if I need to make anything clearer