The diagram below shows a portion a plastic fibre cable, which allows light waves to pass through. The path of the light wave can be modelled by a trigonometric function.

1. Determine the amplitude from the diameter of the cable

$\text{Amplitude} = \frac{\text{Diameter}}{2} = \frac{4}{2} = 2 \text{ mm}$

1. Identify the number of cycles in the diagram and calculate the total length

$\text{Number of cycles} = 3$
$\text{Length} = 3 \times 500 = 1500\text{ nm}$

1. Identify the amplitude and y-intercept to eliminate options

From part (a), the amplitude is $2$. The graph starts at $(0,0)$, so when $x=0$, $y=0$.
For $y = 2\cos(\pi x)$, when $x=0$, $y = 2\cos(0) = 2 \neq 0$. Thus, it is not a cosine function.

2. Use the period to determine the coefficient of x

From part (b), the period is $500$ nm.
For a sine function $y = a\sin(bx)$, the period is $\frac{2\pi}{b}$.
$\frac{2\pi}{b} = 500 \implies b = \frac{2\pi}{500} = \frac{\pi}{250}$.

3. Conclude the correct model

Therefore, the suitable model is $y = 2\sin\left(\frac{\pi}{250}x\right)$.