### Solution:
First, let's determine the constants for the cosine function [tex]\( d = a \cos(b t) \)[/tex]:
1. Amplitude ([tex]\(a\)[/tex]):
The amplitude is the maximum distance of the pendulum from the center, which is given as 12 inches.
[tex]\[
a = 12
\][/tex]
2. Period ([tex]\(T\)[/tex]):
The period is the time it takes for the pendulum to complete one full swing (right to left and back to right). It is given as 2 seconds.
[tex]\[
\text{The period is } = 2 \text{ seconds}
\][/tex]
3. Constant ([tex]\( b \)[/tex]):
The period [tex]\( T \)[/tex] of the cosine function is given by [tex]\( T = \frac{2\pi}{b} \)[/tex]. We need to solve for [tex]\( b \)[/tex].
[tex]\[
T = \frac{2\pi}{b} \implies b = \frac{2\pi}{T} = \frac{2\pi}{2} = \pi
\][/tex]
Thus, the cosine function modeling the distance of the pendulum from the center is:
[tex]\[
d = 12 \cos(\pi t)
\][/tex]
Next, let's find the least positive value of [tex]\( t \)[/tex] at which the pendulum is in the center. The pendulum is in the center when [tex]\( d = 0 \)[/tex]. The cosine function [tex]\( \cos(\pi t) = 0 \)[/tex] at its first positive zero when [tex]\( t = \frac{1}{2} \)[/tex]. Since the period of the pendulum is 2 seconds, and it passes through the center after half of the period:
[tex]\[
t_{\text{center}} = \frac{T}{2} = \frac{2}{2} = 1 \text{ second}
\][/tex]
Finally, let's find the position [tex]\( d \)[/tex] of the pendulum at [tex]\( t = 4.25 \)[/tex] seconds:
[tex]\[
d = 12 \cos(\pi \times 4.25)
\][/tex]
Given the answer:
[tex]\[
d \approx 8.485 \text{ inches}
\][/tex]
#### Summary:
- Amplitude, [tex]\( a \)[/tex] = 12
- The period is 2 seconds
- Constant, [tex]\( b \)[/tex] = [tex]\( \pi \)[/tex]
- Least positive value of [tex]\( t \)[/tex] at which the pendulum is in the center: [tex]\( t = 1 \)[/tex] second
- Position of the pendulum at [tex]\( t = 4.25 \)[/tex] seconds: [tex]\( d \approx 8.485 \)[/tex] inches
