Two simple harmonic motions are represented by `y_(1)=5 [sin 2 pi t + sqrt(3)cos 2 pi t] and y_(2) = 5 sin (2pit+(pi)/(4))`
The ratio of their amplitudes is

Options

A. `1 : 1`
B. `2 : 1`
C. `1 : 3`
D. `sqrt(3) : 1`

Explanation

To find the ratio of the amplitudes of the two simple harmonic motions given by the equations \( y_1 = 5 \left( \sin(2\pi t) + \sqrt{3} \cos(2\pi t) \right) \) and \( y_2 = 5 \sin \left( 2\pi t + \frac{\pi}{4} \right) \), we will follow these steps:

### Step 1: Rewrite \( y_1 \) in a standard form
The first equation can be rewritten using the sine addition formula. We have:

\[ y_1 = 5 \left( \sin(2\pi t) + \sqrt{3} \cos(2\pi t) \right) \]

We can factor out the amplitude from the sine and cosine terms. The amplitude \( A_1 \) can be found using the formula:

\[ A_1 = \sqrt{(5)^2 + (\sqrt{3} \cdot 5)^2} \]

Calculating this gives:

\[ A_1 = \sqrt{25 + 15} = \sqrt{40} = 2\sqrt{10} \]

### Step 2: Identify the amplitude of \( y_2 \)
The second equation is already in a standard form:

\[ y_2 = 5 \sin \left( 2\pi t + \frac{\pi}{4} \right) \]

Here, the amplitude \( A_2 \) is simply the coefficient of the sine function:

\[ A_2 = 5 \]

### Step 3: Calculate the ratio of the amplitudes
Now we can find the ratio of the amplitudes \( A_1 \) and \( A_2 \):

\[ \text{Ratio} = \frac{A_1}{A_2} = \frac{2\sqrt{10}}{5} \]

### Step 4: Simplify the ratio
To express the ratio in a simpler form, we can write:

\[ \text{Ratio} = \frac{2\sqrt{10}}{5} \approx 1.2649 \]

However, if we want the ratio in a more standard form, we can express it as:

\[ \text{Ratio} = \frac{2\sqrt{10}}{5} : 1 \]

### Final Answer
Thus, the ratio of their amplitudes is:

\[ \frac{2\sqrt{10}}{5} : 1 \]