The displacement of an elastic wave is given by the function `y=3 sin omega t +4 cos omegat .`
where y is in cm and t is in second. Calculate the resultant amplitude.

Options

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

Explanation

To find the resultant amplitude of the elastic wave described by the function \( y = 3 \sin(\omega t) + 4 \cos(\omega t) \), we can follow these steps:

### Step 1: Identify the coefficients
The given wave function can be compared to the general form of a wave equation:
\[ y = a \sin(\omega t) + b \cos(\omega t) \]
From the equation, we can identify:
- \( a = 3 \)
- \( b = 4 \)

### Step 2: Use the formula for resultant amplitude
The resultant amplitude \( R \) of a wave described by the equation \( y = a \sin(\omega t) + b \cos(\omega t) \) is given by the formula:
\[ R = \sqrt{a^2 + b^2} \]

### Step 3: Calculate \( a^2 \) and \( b^2 \)
Now we calculate \( a^2 \) and \( b^2 \):
\[ a^2 = 3^2 = 9 \]
\[ b^2 = 4^2 = 16 \]

### Step 4: Substitute into the formula
Now substitute these values into the formula for \( R \):
\[ R = \sqrt{9 + 16} \]

### Step 5: Perform the addition
Calculate the sum:
\[ R = \sqrt{25} \]

### Step 6: Calculate the square root
Finally, calculate the square root:
\[ R = 5 \text{ cm} \]

Thus, the resultant amplitude of the elastic wave is \( 5 \) cm.

### Final Answer
The resultant amplitude is \( 5 \) cm.

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