We have to find the parameters of the model:
![N(t)=A\sin[B(t+C)]+D](https://img.qammunity.org/2023/formulas/mathematics/college/6s724oghfs3xbb0dijpi25oxarf4iqevv0.png)
where N is the number of sunspots and t is the time in years.
(1) The period is 11 years.
This affects the horizontal stretch of the function.
This can be related to the parameter B as:

(2) The maximum number of spots is 300 and the minimum is 0.
This is related to the the parameters A and D, which control the vertical behaviour of the function.
The maximum number of spots happens when the sine function has a value of 1, so we can write:

The minimum number of spots happens when the sine function has a value of -1.
We then can write:

We then can find A and D as:

(3) The remaining parameter, C, will allow us to shift the phase the of the function.
We know that N(0) = 0.
This happens when the sine function is equal to -1.
Then, we can write:
![\begin{gathered} \sin[B(t+C)]=-1 \\ \sin[(2\pi)/(11)(0+C)]=-1 \\ (2\pi)/(11)(C)=(3)/(2)\pi \\ C=(11)/(2)\cdot(3)/(2) \\ C=(33)/(4) \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/dtn3globrb7h2gud83s0fd71893koelo88.png)
We then can write the model as:
![N(t)=150\sin[(2\pi)/(11)(t+(33)/(4))]+150](https://img.qammunity.org/2023/formulas/mathematics/college/u2591ofuvy119vbekw6wzob6pnmgxdl7ec.png)
We can check the accuracy as:
Answer: N(t) = 150*sin(2π/11*(t+33/4))+150
A = 150, B = 2π/11, C = 33/4, D = 150