The displacement of a particle varies according to the relation `x=4 (cos pit+ sinpit)`. The amplitude of the particle is.

Options

A. `8` units
B. `2` units
C. `4`sunits
D. `4sqrt(2)` units

Explanation

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

### Step 1: Identify the Displacement Equation
The displacement of the particle is given as:
\[ x = 4 \cos(\pi t) + \sin(\pi t) \]

### Step 2: Rewrite the Equation
We want to express this equation in the form \( x = A \sin(\omega t + \phi) \) or \( x = A \cos(\omega t + \phi) \). To do this, we can factor out a common term.

### Step 3: Factor Out the Amplitude
To combine the cosine and sine terms, we can use the identity:
\[ R \cos(\theta) + R \sin(\theta) = R \sqrt{a^2 + b^2} \sin(\theta + \phi) \]
where \( R = \sqrt{a^2 + b^2} \) and \( \tan(\phi) = \frac{b}{a} \).

Here, we have:
- \( a = 4 \) (coefficient of \( \cos(\pi t) \))
- \( b = 1 \) (coefficient of \( \sin(\pi t) \))

### Step 4: Calculate the Amplitude
Now we can calculate the amplitude \( A \):
\[ A = \sqrt{(4)^2 + (1)^2} = \sqrt{16 + 1} = \sqrt{17} \]

### Step 5: Final Result
Thus, the amplitude of the particle is:
\[ A = \sqrt{17} \]

### Summary
The amplitude of the particle is \( \sqrt{17} \).

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