Two simple harmonic motions are represented by the equations
`y_(1) = 8 sin(pi)/(4)(10t + 2),y_(2) = 6(sin 5pit + sqrt(3) cos 5omegat)`
IF the ratio of amplitudes of `y_(1) and y_(2)` is `alpha` and their respective ratio of time periods is `beta`, then find `beta//alpha`.

Explanation

To solve the problem, we need to find the ratio of amplitudes (α) and the ratio of time periods (β) of the two simple harmonic motions given by the equations:

1. \( y_1 = 8 \sin\left(\frac{\pi}{4}(10t + 2)\right) \)
2. \( y_2 = 6\left(\sin(5\pi t) + \sqrt{3} \cos(5\pi t)\right) \)

### Step 1: Identify the Amplitudes

From the equation of \( y_1 \):
- The amplitude \( A_1 = 8 \).

For \( y_2 \), we need to rewrite it in the standard form \( A \sin(\omega t + \phi) \):
- We can express \( y_2 \) as:
\[ y_2 = 6\left(\sin(5\pi t) + \sqrt{3} \cos(5\pi t)\right) \]
Using the formula \( R = \sqrt{a^2 + b^2} \) where \( a = 6 \) and \( b = 6\sqrt{3} \), we can find the amplitude:
\[ R = 6\sqrt{1^2 + \left(\sqrt{3}\right)^2} = 6\sqrt{1 + 3} = 6\sqrt{4} = 12 \]
Thus, the amplitude \( A_2 = 12 \).

### Step 2: Calculate the Ratio of Amplitudes (α)

Now we can find the ratio of the amplitudes:
\[ \alpha = \frac{A_1}{A_2} = \frac{8}{12} = \frac{2}{3} \]

### Step 3: Identify the Angular Frequencies and Time Periods

Next, we need to find the time periods of both motions.

For \( y_1 \):
- The angular frequency \( \omega_1 \) can be found from the equation:
\[ \frac{\pi}{4} \cdot 10 = \omega_1 \implies \omega_1 = \frac{10\pi}{4} = \frac{5\pi}{2} \]
- The time period \( T_1 \) is given by:
\[ T_1 = \frac{2\pi}{\omega_1} = \frac{2\pi}{\frac{5\pi}{2}} = \frac{4}{5} \]

For \( y_2 \):
- The angular frequency \( \omega_2 = 5\pi \).
- The time period \( T_2 \) is given by:
\[ T_2 = \frac{2\pi}{\omega_2} = \frac{2\pi}{5\pi} = \frac{2}{5} \]

### Step 4: Calculate the Ratio of Time Periods (β)

Now we can find the ratio of the time periods:
\[ \beta = \frac{T_1}{T_2} = \frac{\frac{4}{5}}{\frac{2}{5}} = \frac{4}{2} = 2 \]

### Step 5: Calculate the Ratio β/α

Finally, we need to find the ratio \( \frac{\beta}{\alpha} \):
\[ \frac{\beta}{\alpha} = \frac{2}{\frac{2}{3}} = 2 \times \frac{3}{2} = 3 \]

### Final Answer

Thus, the value of \( \frac{\beta}{\alpha} \) is \( 3 \).

---