A unit step signal, u(t) is defined as: u(t < 0) = 0 and u(t > 0)=1.

If x(t) = u(10-2t), which one of the following is true?

A.  x(t>5)=1,x(t<5)=0
B.  x(t<-5)=1,x(t>-5)=0
C.  x(t<5)=1,x(t>5)=0
D.  x(t<-5)=1,x(t>-5)=0

The given signal is \( x(t) = u(10 - 2t) \), where \( u(t) \) is the unit step function. The unit step function \( u(t) \) is defined as:

\[
u(t) = 
\begin{cases}
0 & \text{if } t < 0 \\
1 & \text{if } t \geq 0
\end{cases}
\]

We need to determine the values of \( t \) where \( x(t) \) transitions from 0 to 1 by solving the inequality \( 10 - 2t = 0 \).

### Step 1: Find the critical point
Set the argument of the step function to zero to find the critical time where the step occurs:
\[
10 - 2t = 0
\]
\[
2t = 10
\]
\[
t = 5
\]

### Step 2: Analyze behavior of \( x(t) \)

- For \( t < 5 \):
  \[
  10 - 2t > 0 \quad \Rightarrow \quad u(10 - 2t) = 1
  \]
  So, \( x(t) = 1 \) when \( t < 5 \).

- For \( t > 5 \):
  \[
  10 - 2t < 0 \quad \Rightarrow \quad u(10 - 2t) = 0
  \]
  So, \( x(t) = 0 \) when \( t > 5 \).

### Step 3: Conclusion
The correct description is:
\[
x(t) = 
\begin{cases}
1 & \text{if } t < 5 \\
0 & \text{if } t > 5
\end{cases}
\]

This matches option **C**.

### Final Answer: **C** \( x(t<5)=1, x(t>5)=0 \)

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Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How is the unit step function \( u(t) \) used in signal processing?
2. How would the function \( x(t) \) change if the coefficient of \( t \) were different in \( u(10 - 2t) \)?
3. What is the graphical representation of a unit step function?
4. How do unit step functions combine in Laplace transforms?
5. Can you describe how \( u(t) \) shifts and scales based on the argument inside the function?

**Tip**: In signal processing, the unit step function is often used to model signals that turn on or off at a specific point in time.

A unit step signal, u(t), is defined as: u(t < 0) = 0 and u(t > 0)=1.
If x(t) = u(10-2t), which one of the following is true?

A.  x(t>5)=1, x(t<5)=0
B.  x(t<-5)=1, x(t>-5)=0
C.  x(t<5)=1, x(t>5)=0
D.  x(t<-5)=1, x(t>-5)=0

Explore the solution to the unit step function x(t) = u(10 - 2t) and understand how the signal behaves for t < 5 and t > 5. This problem covers key concepts in signal processing and the behavior of the unit step function in transformations.

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