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 will follow the given wave equation and the conditions provided.

### Given:
The wave equation is:
\[ y = 10 \sin\left(\frac{2\pi}{45}t + \alpha\right) \]

### Step 1: Find the value of \(\alpha\)
At \(t = 0\), the displacement \(y\) is given as 5 cm. We can substitute \(t = 0\) into the wave equation:

\[ y = 10 \sin(\alpha) \]
Setting \(y = 5\):
\[ 5 = 10 \sin(\alpha) \]

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

### Step 3: Determine \(\alpha\)
The value of \(\alpha\) for which \(\sin(\alpha) = \frac{1}{2}\) is:
\[ \alpha = \frac{\pi}{6} \quad \text{(or 30 degrees)} \]

### Step 4: Calculate the total phase at \(t = 7.5\) seconds
The total phase \(\phi\) at time \(t\) is given by:
\[ \phi = \frac{2\pi}{45}t + \alpha \]
Substituting \(t = 7.5\) seconds:
\[ \phi = \frac{2\pi}{45} \times 7.5 + \frac{\pi}{6} \]

### Step 5: Calculate \(\frac{2\pi}{45} \times 7.5\)
Calculating:
\[ \frac{2\pi}{45} \times 7.5 = \frac{2\pi \times 7.5}{45} = \frac{15\pi}{45} = \frac{\pi}{3} \]

### Step 6: Add \(\alpha\) to the phase
Now, substituting \(\alpha = \frac{\pi}{6}\):
\[ \phi = \frac{\pi}{3} + \frac{\pi}{6} \]

### Step 7: Find a common denominator and add
The common denominator for \(\frac{\pi}{3}\) and \(\frac{\pi}{6}\) is 6:
\[ \frac{\pi}{3} = \frac{2\pi}{6} \]
So:
\[ \phi = \frac{2\pi}{6} + \frac{\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2} \]

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

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