Two simple harmonic motions are represented by the equations.
`y_(1)=10"sin"(pi)/(4)(12t+1),y_(2)=5(sin3pt+sqrt(3)cos3pt)` the ratio of their amplitudes is

Options

A. `1:1`
B. `1:2`
C. `3:2`
D. `2:3`

Explanation

To find the ratio of the amplitudes of the two simple harmonic motions represented by the equations \(y_1\) and \(y_2\), we will follow these steps:

### Step 1: Identify the amplitude of \(y_1\)
The equation for \(y_1\) is given as:
\[ y_1 = 10 \sin\left(\frac{\pi}{4}(12t + 1)\right) \]
We can rewrite this equation in the standard form \(y = A \sin(\omega t + \phi)\). Here, the amplitude \(A_1\) is simply the coefficient of the sine function.

**Amplitude of \(y_1\)**:
\[ A_1 = 10 \]

### Step 2: Identify the amplitude of \(y_2\)
The equation for \(y_2\) is given as:
\[ y_2 = 5 \sin(3\pi t) + \sqrt{3} \cos(3\pi t) \]
To find the amplitude \(A_2\), we can express \(y_2\) in the form \(A \sin(\omega t + \phi)\). We will use the formula for the amplitude of a combination of sine and cosine:
\[ A = \sqrt{A^2 + B^2} \]
where \(A\) is the coefficient of the sine term and \(B\) is the coefficient of the cosine term.

In this case, \(A = 5\) and \(B = \sqrt{3}\).

**Calculating the amplitude \(A_2\)**:
\[ A_2 = \sqrt{(5)^2 + (\sqrt{3})^2} = \sqrt{25 + 3} = \sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7} \]

### Step 3: Find the ratio of the amplitudes
Now, we can find the ratio of the amplitudes \(A_1\) and \(A_2\):
\[ \text{Ratio} = \frac{A_1}{A_2} = \frac{10}{2\sqrt{7}} = \frac{5}{\sqrt{7}} \]

### Step 4: Rationalize the denominator (if needed)
To express the ratio in a more standard form, we can rationalize the denominator:
\[ \text{Ratio} = \frac{5\sqrt{7}}{7} \]

Thus, the final answer for the ratio of the amplitudes is:
\[ \frac{5\sqrt{7}}{7} \]