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give manually soved step  by step solution in detail

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### Problem Summary
1. We are given a piecewise function \( f(t) \) represented in the graph.
2. The task has two parts:
   - **Part (a):** Express \( f(t) \) using Heaviside step functions.
   - **Part (b):** Find the Laplace transform \( F(s) = \mathcal{L}\{f(t)\} \).

### Solution

#### Step 1: Analyzing the Function \( f(t) \) from the Graph

From the graph:
- \( f(t) \) starts at \( t = 0 \) and is \( f(0) = 0 \).
- From \( t = 0 \) to \( t = 4 \), the function increases linearly to \( f(4) = 4 \).
- From \( t = 4 \) to \( t = 8 \), \( f(t) \) is constant at 4.
- For \( t > 8 \), \( f(t) \) drops back to 0.

This suggests that \( f(t) \) can be defined in three sections:
1. \( f(t) = \frac{t}{4} \) for \( 0 \leq t < 4 \).
2. \( f(t) = 4 \) for \( 4 \leq t < 8 \).
3. \( f(t) = 0 \) for \( t \geq 8 \).

#### Step 2: Expressing \( f(t) \) Using Heaviside Step Functions

The Heaviside function, \( h(t - a) \), is defined as:
\[
h(t - a) = 
\begin{cases} 
0 & \text{for } t < a \\ 
1 & \text{for } t \geq a 
\end{cases}
\]

We can use this to "turn on" or "turn off" parts of the function at specific points.

1. **First segment (from \( 0 \) to \( 4 \)):** \( f(t) = \frac{t}{4} \) for \( 0 \leq t < 4 \).
   - This part can be written as \( \frac{t}{4} \cdot h(t) \).

2. **Second segment (from \( 4 \) to \( 8 \)):** \( f(t) = 4 \) for \( 4 \leq t < 8 \).
   - We subtract the value at \( t = 4 \) and add it back with a constant to match this section:
   \[
   4 \cdot h(t - 4) - 4 \cdot h(t - 8)
   \]

3. **Third segment (for \( t \geq 8 \)):** The function becomes zero, which we accomplish by subtracting the previous terms after \( t = 8 \).

Thus, we can combine these terms to represent \( f(t) \) as:
\[
f(t) = \frac{t}{4} \cdot h(t) + (4 - \frac{t}{4}) h(t - 4) - 

Give manually solved step-by-step solution in detail for the provided problem involving Heaviside step functions and Laplace transforms.

This solution provides a detailed, step-by-step approach to expressing a piecewise function using Heaviside step functions and finding its Laplace transform. Suitable for college-level students studying engineering or mathematics.

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