Answer:
[tex]f(x)=3\sin\left(\frac{\pi}{12}\left(x-10\right)\right)+13[/tex].
Step-by-step explanation:
Given information:
Period = 24 hr
Maximum = 16 at t=16 hr.
Minimum = 10 at t=4 hr.
The general sin function is
[tex]y=A+\sin(B(x-C))+D[/tex] .... (1)
where, |A| is altitude, [tex]\frac{2\pi}{B}[/tex] is period, C is phase shift and D is midline.
Period is 24 hr.
[tex]24=\dfrac{2\pi}{B}\Rightarrow B=\dfrac{\pi}{12}[/tex]
Altitude is
[tex]A=\dfrac{Maximum-Minimum}{2}=\dfrac{16-10}{2}=3[/tex]
[tex]D=\dfrac{Maximum+Minimum}{2}=\dfrac{16+10}{2}=13[/tex]
The function is minimum at t=4 and maximum at t=16,phase shift is
[tex]C=\dfrac{16+4}{2}=10[/tex]
Substitute these values in equation (1).
[tex]y=3\sin\left(\frac{\pi}{12}\left(x-10\right)\right)+13[/tex]
Therefore, the required function is [tex]f(x)=3\sin\left(\frac{\pi}{12}\left(x-10\right)\right)+13[/tex].
