The problem asks us to find the period of the function $y = 10 \sin(\frac{6\pi}{4}(x - \frac{\pi}{2})) + 25$.

1. Problem Description

The problem asks us to find the period of the function $y = 10 \sin(\frac{6\pi}{4}(x - \frac{\pi}{2})) + 25$.

2. Solution Steps

The general form of a sinusoidal function is $y = A\sin(B(x - C)) + D$, where $A$ is the amplitude, $B$ is related to the period, $C$ is the horizontal shift, and $D$ is the vertical shift. The period $P$ of the function is given by the formula:

$P = \frac{2\pi}{|B|}$

In the given function, $y = 10 \sin(\frac{6\pi}{4}(x - \frac{\pi}{2})) + 25$, we have $B = \frac{6\pi}{4}$.

Therefore, the period is:

$P = \frac{2\pi}{|\frac{6\pi}{4}|} = \frac{2\pi}{\frac{6\pi}{4}} = 2\pi \cdot \frac{4}{6\pi} = \frac{8\pi}{6\pi} = \frac{4}{3}$.

3. Final Answer

The period of the function is $\frac{4}{3}$.