Answer:
[tex]25.4\,days \leq t \leq 34.6\,days[/tex]
Step-by-step explanation:
Let consider the following model:
[tex]c(t) = A \cdot \sin \left(\frac{2\pi \cdot t}{T} \right) + \bar c[/tex]
The average is given by the following formula:
[tex]\bar c = \frac{c_{min}+c_{max}}{2}[/tex]
The maximum value is:
[tex]c_{max} = 2 \cdot \bar c - c_{min}[/tex]
[tex]c_{max} = 2\cdot (8\,in) - 4\,in[/tex]
[tex]c_{max} = 12\,in[/tex]
Amplitude is:
[tex]A = 12\,in - 8\,in[/tex]
[tex]A = 4\,in[/tex]
The sine function has a periodicity of [tex]2\pi[/tex], where is minimum is reached at [tex]\theta = \frac{3\pi}{2}[/tex], when t = 30. The period of the cycle is:
[tex]T = \frac{2\pi}{\frac{3}{2}\pi }\cdot (30)[/tex]
[tex]T = 40[/tex]
The complete expression is:
[tex]c(t) = 8\,in + 4\,in \cdot \sin \left(\frac{2\pi}{40}\cdot t \right)[/tex]
The times associated with [tex]c = 5\,in[/tex] are, respectively:
[tex]8\,in + 4\,in \cdot \sin \left(\frac{2\pi}{40}\cdot t \right) = 5\,in[/tex]
[tex]4\,in \cdot \sin \left(\frac{2\pi}{40}\cdot t \right) = -3\,in[/tex]
[tex]\sin \left(\frac{2\pi}{40}\cdot t \right) = -0.75[/tex]
[tex]t = \frac{40}{2\pi}\cdot \sin^{-1} (-0.75)[/tex]
Instants are, respectively:
[tex]t_{1} \approx 25.4\,days[/tex]
[tex]t_{2} \approx 34.6\,days[/tex]
Period is:
[tex]25.4\,days \leq t \leq 34.6\,days[/tex]
