To determine the initial position of the paddle, we need to evaluate the given function [tex]\( f(t) = 5.25 \cos \left(\frac{\pi}{16} t - \pi \right) \)[/tex] at the initial time [tex]\( t = 0 \)[/tex].
1. Substitute [tex]\( t = 0 \)[/tex] into the function [tex]\( f(t) \)[/tex]:
[tex]\[
f(0) = 5.25 \cos \left(\frac{\pi}{16} \cdot 0 - \pi \right)
\][/tex]
2. Simplify the argument of the cosine function:
[tex]\[
\frac{\pi}{16} \cdot 0 - \pi = -\pi
\][/tex]
3. Evaluate the cosine of [tex]\(-\pi\)[/tex]:
The cosine function is periodic and even, so:
[tex]\[
\cos(-\pi) = \cos(\pi)
\][/tex]
We know that:
[tex]\[
\cos(\pi) = -1
\][/tex]
4. Substitute [tex]\(\cos(\pi) = -1\)[/tex] back into the function:
[tex]\[
f(0) = 5.25 \cdot (-1) = -5.25
\][/tex]
5. Interpret the result:
Since the function value at [tex]\( t = 0 \)[/tex] is [tex]\(-5.25\)[/tex], this indicates that the paddle is below the waterline.
Therefore, the initial position of the paddle is [tex]\( 5.25 \)[/tex] feet below the waterline. The correct answer is:
[tex]\[ \boxed{5.25 \text{ ft below the waterline}} \][/tex]
