The equation of a wave is given by
`y=10sin((2pi)/45t+alpha)`
If the displacement is 5 cm at t = 0, then the total phase at t = 7.5 s is

Options

A. `pi/3`
B. `pi/2`
C. `pi/6`
D. `pi`

Explanation

To solve the problem step by step, we start with the wave equation and the given conditions.

### Step 1: Write down the wave equation
The equation of the wave is given by:
\[ y = 10 \sin\left(\frac{2\pi}{45}t + \alpha\right) \]

### Step 2: Use the initial condition to find α
We know that at \( t = 0 \), the displacement \( y \) is 5 cm. Therefore, we can substitute \( t = 0 \) into the wave equation:
\[ 5 = 10 \sin\left(\frac{2\pi}{45} \cdot 0 + \alpha\right) \]
This simplifies to:
\[ 5 = 10 \sin(\alpha) \]

### Step 3: Solve for sin(α)
Dividing both sides by 10 gives:
\[ \sin(\alpha) = \frac{5}{10} = \frac{1}{2} \]

### Step 4: Determine α
The angle \( \alpha \) for which \( \sin(\alpha) = \frac{1}{2} \) is:
\[ \alpha = \frac{\pi}{6} \text{ (or 30 degrees)} \]

### Step 5: Calculate the total phase at \( t = 7.5 \) s
The total phase \( \phi \) at time \( t \) can be calculated using the wave equation:
\[ \phi = \frac{2\pi}{45}t + \alpha \]
Substituting \( t = 7.5 \) s and \( \alpha = \frac{\pi}{6} \):
\[ \phi = \frac{2\pi}{45} \cdot 7.5 + \frac{\pi}{6} \]

### Step 6: Calculate \( \frac{2\pi}{45} \cdot 7.5 \)
Calculating \( \frac{2\pi}{45} \cdot 7.5 \):
\[ \frac{2\pi \cdot 7.5}{45} = \frac{15\pi}{45} = \frac{\pi}{3} \]

### Step 7: Add the phases
Now, we add \( \frac{\pi}{3} \) and \( \frac{\pi}{6} \):
To add these fractions, we need a common denominator:
\[ \frac{\pi}{3} = \frac{2\pi}{6} \]
Thus:
\[ \phi = \frac{2\pi}{6} + \frac{\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2} \]

### Final Answer
The total phase at \( t = 7.5 \) s is:
\[ \phi = \frac{\pi}{2} \]

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