The displacement of a particle is given by ` x = 3 sin ( 5 pi t) + 4 cos ( 5 pi t)`. The amplitude of particle is

Options

A. `3`
B. `4`
C. `5`
D. `7`

Explanation

To find the amplitude of the particle given the displacement equation \( x = 3 \sin(5 \pi t) + 4 \cos(5 \pi t) \), we can follow these steps:

### Step 1: Identify the components
The displacement equation consists of two components: \( 3 \sin(5 \pi t) \) and \( 4 \cos(5 \pi t) \). Here, the coefficients (3 and 4) represent the amplitudes of the sine and cosine components, respectively.

### Step 2: Convert to a single sine function
We can express the combination of sine and cosine functions into a single sine function using the formula:
\[ R = \sqrt{A^2 + B^2} \]
where \( A \) is the coefficient of the sine function and \( B \) is the coefficient of the cosine function.

### Step 3: Calculate the resultant amplitude
In our case:
- \( A = 3 \)
- \( B = 4 \)

Now, we calculate the resultant amplitude \( R \):
\[ R = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 4: Conclusion
The amplitude of the particle is \( 5 \).

### Final Answer
The amplitude of the particle is \( 5 \).

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