The amplitude of a wave represented by the equation `y=3sin(5x-0.5t)+4cos(5x-0.5t)`, is

Options

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

Answer

To find the amplitude of the wave represented by the equation \( y = 3\sin(5x - 0.5t) + 4\cos(5x - 0.5t) \), we can follow these steps:

### Step 1: Identify the components of the wave
The given wave equation consists of two components:
- The sine component: \( 3\sin(5x - 0.5t) \)
- The cosine component: \( 4\cos(5x - 0.5t) \)

### Step 2: Convert the cosine term to sine
To combine these two components, we can convert the cosine term into a sine term. We know that:
\[ \cos(\theta) = \sin\left(\theta + \frac{\pi}{2}\right) \]
Thus, we can rewrite the cosine term:
\[ 4\cos(5x - 0.5t) = 4\sin\left(5x - 0.5t + \frac{\pi}{2}\right) \]
This means we can express the wave equation as:
\[ y = 3\sin(5x - 0.5t) + 4\sin\left(5x - 0.5t + \frac{\pi}{2}\right) \]

### Step 3: Use phasor analysis
Now we can apply phasor analysis. We treat the sine components as vectors (phasors):
- The first phasor has a magnitude of 3 (along the horizontal axis).
- The second phasor has a magnitude of 4 and is at a phase angle of \( \frac{\pi}{2} \) (along the vertical axis).

### Step 4: Calculate the resultant amplitude
To find the resultant amplitude, we can use the Pythagorean theorem, since the two phasors are perpendicular to each other:
\[ A = \sqrt{(3)^2 + (4)^2} \]
Calculating this gives:
\[ A = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Conclusion
The amplitude of the wave represented by the equation \( y = 3\sin(5x - 0.5t) + 4\cos(5x - 0.5t) \) is 5 units.