To determine how long it takes for the waterwheel to complete one turn, we need to analyze the given equation:
[tex]\[ h = 15 \cos \left(\frac{\pi}{20} t \right) \][/tex]
In this equation, [tex]\(h\)[/tex] represents the height of the piece of cloth tied to the waterwheel, and [tex]\(t\)[/tex] represents time in seconds. The function inside the cosine, [tex]\(\frac{\pi}{20} t\)[/tex], determines the angular position of the waterwheel as it rotates over time.
To complete one full turn, the argument of the cosine function must change by [tex]\(2\pi\)[/tex] radians, since [tex]\(2\pi\)[/tex] radians is one complete cycle in a trigonometric function.
1. Set up the equation for one full turn:
[tex]\[ \frac{\pi}{20} t = 2\pi \][/tex]
2. Solve for [tex]\(t\)[/tex]:
[tex]\[
\begin{align*}
\frac{\pi}{20} t &= 2\pi \\
t &= \frac{2\pi}{\frac{\pi}{20}} \\
t &= 2\pi \times \frac{20}{\pi} \\
t &= 2\pi \times \frac{20}{\pi} \\
t &= 2 \times 20 \\
t &= 40 \, \text{seconds}
\end{align*}
\][/tex]
Therefore, it takes the waterwheel 40 seconds to complete one full turn.
The correct answer is: [tex]\( \boxed{40 \text{ seconds}} \)[/tex]
