Two simple harmonic motions are represented by the equations
`y_(1) = 10 sin(3pit + pi//4) and y_(2) = 5(sin 3pit + sqrt(3)cos 3pit)` their amplitude are in the ratio of ………… .

Options

A. `1:2`
B. `2:1`
C. `1:1`
D. None of these

Explanation

To solve the problem, we need to determine the amplitudes of the two simple harmonic motions represented by the equations:

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

### Step 1: Identify the amplitude of the first equation

The amplitude of the first equation \( y_1 \) can be directly read from the equation:

\[ A_1 = 10 \]

### Step 2: Rewrite the second equation in standard form

The second equation \( y_2 = 5(\sin(3\pi t) + \sqrt{3}\cos(3\pi t)) \) needs to be rewritten in the form \( R \sin(3\pi t + \phi) \) to find its amplitude.

To do this, we can use the formula for the resultant amplitude when combining sine and cosine:

\[ R = \sqrt{A^2 + B^2} \]

where \( A \) is the coefficient of \( \sin(3\pi t) \) and \( B \) is the coefficient of \( \cos(3\pi t) \).

Here, \( A = 5 \) and \( B = 5\sqrt{3} \).

### Step 3: Calculate the resultant amplitude \( R \)

Now, we can calculate \( R \):

\[ R = \sqrt{(5)^2 + (5\sqrt{3})^2} \]

Calculating each term:

\[ (5)^2 = 25 \]
\[ (5\sqrt{3})^2 = 25 \cdot 3 = 75 \]

Now, add these:

\[ R = \sqrt{25 + 75} = \sqrt{100} = 10 \]

### Step 4: Identify the amplitude of the second equation

Thus, the amplitude of the second equation \( y_2 \) is:

\[ A_2 = 10 \]

### Step 5: Find the ratio of the amplitudes

Now that we have both amplitudes:

\[ A_1 = 10, \quad A_2 = 10 \]

The ratio of the amplitudes \( \frac{A_1}{A_2} \) is:

\[ \text{Ratio} = \frac{10}{10} = 1 \]

Thus, the ratio of the amplitudes is:

\[ 1:1 \]

### Final Answer

The ratio of the amplitudes is \( 1:1 \).

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