A pulse is started at a time `t = 0` along the `+x` directions an a long, taut string. The shaot of the puise at `t = 0` is given by funcation `y` with
`y = {{:((x)/(4)+1fo r-4ltxle0),(-x+1fo r0ltxlt1),("0 otherwise"):}`
here `y` and `x` are in centimeters. The linear mass density of the string is `50 g//m` and it is under a tension of `5N`,
the shape of the string is drawn at `t = 0` and lthe area of the pulse elclosed by the string and the `x-`string is measured. It will be equal to `

Options

A. `2 cm^(2)`
B. `2.5 cm^(2)`
C. `4 cm^(2)`
D. `5 cm^(2)`

Explanation

To solve the problem, we need to find the area enclosed by the pulse shape described by the function \( y \) at \( t = 0 \). The function is defined piecewise, and we will analyze it step by step.

### Step 1: Understand the function
The function \( y \) is defined as follows:
- \( y = \frac{x}{4} + 1 \) for \( -4 < x < 0 \)
- \( y = -x + 1 \) for \( 0 < x < 1 \)
- \( y = 0 \) otherwise

### Step 2: Sketch the graph
To visualize the pulse:
1. For \( -4 < x < 0 \):
- At \( x = -4 \): \( y = \frac{-4}{4} + 1 = 0 \)
- At \( x = 0 \): \( y = \frac{0}{4} + 1 = 1 \)
- This gives us a straight line from the point (-4, 0) to (0, 1).

2. For \( 0 < x < 1 \):
- At \( x = 0 \): \( y = -0 + 1 = 1 \)
- At \( x = 1 \): \( y = -1 + 1 = 0 \)
- This gives us a straight line from the point (0, 1) to (1, 0).

### Step 3: Identify the shape of the pulse
The graph consists of two linear segments forming a triangle:
- The base of the triangle runs from \( x = -4 \) to \( x = 1 \) (length = 5 units).
- The height of the triangle is from \( y = 0 \) to \( y = 1 \) (height = 1 unit).

### Step 4: Calculate the area of the triangle
The area \( A \) of a triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the values we found:
\[ A = \frac{1}{2} \times 5 \times 1 = \frac{5}{2} = 2.5 \, \text{cm}^2 \]

### Step 5: Conclusion
The area of the pulse enclosed by the string and the x-axis at \( t = 0 \) is \( 2.5 \, \text{cm}^2 \).

---