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For part (d), we need to determine the x-coordinate of the right end of the rod at point \( B \) as a function of time \( t \).

Given information:
1. The rod rotates with point \( A \) on a circle of radius \( 4 \) cm, with center at the origin, and point \( A \) has a circular motion with angular velocity \( \omega = 4\pi \) radians per second.
2. The rod length \( AB = 8 \) cm, with \( B \) moving along the x-axis.

To express the x-coordinate of point \( B \) in terms of \( t \):

1. **Position of Point \( A \):** 
   - From part (c), the coordinates of point \( A \) are:
     \[
     x_A = 4 \cos(4\pi t)
     \]
     \[
     y_A = 4 \sin(4\pi t)
     \]

2. **Determining the x-coordinate of \( B \):**
   - The x-coordinate of \( B \) can be calculated by extending from \( A \) by the length of the rod \( AB = 8 \) cm along the horizontal direction.
   - Since point \( B \) lies horizontally along the x-axis relative to \( A \), its x-coordinate is given by:
     \[
     x_B = x_A + 8 \cos(4\pi t)
     \]
   - This becomes:
     \[
     x_B = 4 \cos(4\pi t) + 8 \cos(4\pi t) = (4 + 8) \cos(4\pi t) = 12 \cos(4\pi t)
     \]

Thus, the x-coordinate of point \( B \) as a function of \( t \) is:
\[
x_B = 12 \cos(4\pi t)
\]

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Would you like more details on this solution or have any questions?

### Related Questions:
1. How would the answer change if the wheel rotated at a different speed?
2. What is the y-coordinate of point \( B \), and why does it remain zero?
3. How would you calculate the velocity of point \( B \) at any time \( t \)?
4. How can we find the angular position of point \( A \) as a function of \( t \)?
5. What happens to point \( B \)’s motion if the rod length \( AB \) changes?

**Tip:** To determine the x-coordinate of a point moving in relation to another, consider both the length and the angle of the connecting segment.

A rod is attached to a rotating point A on a wheel of radius 4 cm, moving counterclockwise at 2.0 revolutions per second. Calculate the x-coordinate of point B as a function of time t.

Find the x-coordinate of the right end of a rod as a function of time as it rotates with a wheel. This problem involves concepts of circular motion, angular velocity, and trigonometry.

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