Find the time `t_0` when x-coordinate of the particle is zero.

Explanation

To find the time \( t_0 \) when the x-coordinate of the particle is zero, we can follow these steps:

### Step 1: Understand the Problem
We need to determine the time \( t_0 \) when the position \( x \) of a particle is equal to zero. We have a graph that shows the relationship between time \( t \) and position \( x \).

### Step 2: Identify Points on the Graph
From the video transcript, we have the following points:
- At \( t = 2 \) seconds, \( x = 10 \) meters
- At \( t = 4 \) seconds, \( x = 18 \) meters
- At \( t = 8 \) seconds, \( x = 2 \) meters
- At \( t = 12 \) seconds, \( x = -30 \) meters

### Step 3: Determine the Relevant Segment
We need to find the time when \( x = 0 \). Observing the points, we see that the particle moves from \( x = 2 \) meters at \( t = 8 \) seconds to \( x = -30 \) meters at \( t = 12 \) seconds. This indicates that the particle crosses \( x = 0 \) between these two times.

### Step 4: Calculate the Slope
We can find the slope of the line segment between \( t = 8 \) seconds and \( t = 12 \) seconds. The change in position (\( \Delta x \)) and change in time (\( \Delta t \)) is:
- \( \Delta x = -30 - 2 = -32 \) meters
- \( \Delta t = 12 - 8 = 4 \) seconds

The slope \( m \) is given by:
\[ m = \frac{\Delta x}{\Delta t} = \frac{-32}{4} = -8 \]

### Step 5: Write the Equation of the Line
Using the point-slope form of the equation of a line, we can write:
\[ x = mt + c \]
Substituting \( m = -8 \):
\[ x = -8t + c \]

### Step 6: Find the y-intercept \( c \)
We can use one of the known points to find \( c \). Let's use the point \( (8, 2) \):
\[ 2 = -8(8) + c \]
\[ 2 = -64 + c \implies c = 66 \]
Thus, the equation becomes:
\[ x = -8t + 66 \]

### Step 7: Set \( x = 0 \) and Solve for \( t_0 \)
To find \( t_0 \) when \( x = 0 \):
\[ 0 = -8t_0 + 66 \]
Rearranging gives:
\[ 8t_0 = 66 \implies t_0 = \frac{66}{8} = 8.25 \text{ seconds} \]

### Final Answer
The time \( t_0 \) when the x-coordinate of the particle is zero is \( t_0 = 8.25 \) seconds.

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