Question

During a 90-day monsoon season, daily rainfall can be modeled by sinusoidal functions. A low of 4 inches of rainfall was recorded on day 30, and overall the average daily rainfall was 8 inches. During what period was daily rainfall less than 5 inches?

Answer 1

Answer:

$25.4\,days \leq t \leq 34.6\,days$

Step-by-step explanation:

Let consider the following model:

$c(t) = A \cdot \sin \left(\frac{2\pi \cdot t}{T} \right) + \bar c$

The average is given by the following formula:

$\bar c = \frac{c_{min}+c_{max}}{2}$

The maximum value is:

$c_{max} = 2 \cdot \bar c - c_{min}$

$c_{max} = 2\cdot (8\,in) - 4\,in$

$c_{max} = 12\,in$

Amplitude is:

$A = 12\,in - 8\,in$

$A = 4\,in$

The sine function has a periodicity of $2\pi$, where is minimum is reached at $\theta = \frac{3\pi}{2}$, when t = 30.  The period of the cycle is:

$T = \frac{2\pi}{\frac{3}{2}\pi }\cdot (30)$

$T = 40$

The complete expression is:

$c(t) = 8\,in + 4\,in \cdot \sin \left(\frac{2\pi}{40}\cdot t \right)$

The times associated with $c = 5\,in$ are, respectively:

$8\,in + 4\,in \cdot \sin \left(\frac{2\pi}{40}\cdot t \right) = 5\,in$

$4\,in \cdot \sin \left(\frac{2\pi}{40}\cdot t \right) = -3\,in$

$\sin \left(\frac{2\pi}{40}\cdot t \right) = -0.75$

$t = \frac{40}{2\pi}\cdot \sin^{-1} (-0.75)$

Instants are, respectively:

$t_{1} \approx 25.4\,days$

$t_{2} \approx 34.6\,days$

Period is:

$25.4\,days \leq t \leq 34.6\,days$