For the given wave equation y = 10 π((x)/(4) + (t)/(2)) , the...

The given wave equation is $y = 10\cos\pi\left(\dfrac{x}{4} + \dfrac{t}{2}\right)$

Rewriting the equation, we get $y = 10\cos\left(\dfrac{\pi}{4}x + \dfrac{\pi}{2}t\right)$

Comparing with the standard wave equation $y = A\cos(kx + \omega t)$, we get:

Angular frequency $\omega = \dfrac{\pi}{2}\ \text{rad/s}$

Wave number $k = \dfrac{\pi}{4}\ \text{m}^{-1}$

Frequency $f = \dfrac{\omega}{2\pi} = \dfrac{\pi/2}{2\pi} = 0.25\ \text{Hz}$

Wavelength $\lambda = \dfrac{2\pi}{k} = \dfrac{2\pi}{\pi/4} = 8\ \text{m}$

Wave velocity $v = f\lambda = 0.25 \times 8 = 2\ \text{m/s} = 200\ \text{cm/s}$

Thus, the only correct statement is frequency $= 0.25\ \text{Hz}$.

Answer: frequency $= 0.25\ \text{Hz}$