To determine how long it takes for the waterwheel to complete one turn, we need to find the period of the given cosine function. The period of a trigonometric function describes how long it takes for the function to complete one full cycle.
The given equation for the height [tex]\(h\)[/tex] in feet of the piece of cloth tied to the waterwheel, as a function of time [tex]\(t\)[/tex] in seconds, is:
[tex]\[ h = 15 \cos \left(\frac{\pi}{20} t\right) \][/tex]
Step-by-Step Solution:
1. Identify the standard form of a cosine function:
The standard form of a cosine function is [tex]\( h = A \cos (Bt) \)[/tex], where [tex]\( A \)[/tex] is the amplitude and [tex]\( B \)[/tex] affects the period of the function.
2. Determine [tex]\( B \)[/tex] from the given function:
The given function is [tex]\( h = 15 \cos \left(\frac{\pi}{20} t\right) \)[/tex].
Here, [tex]\( B = \frac{\pi}{20} \)[/tex].
3. Calculate the period of the function:
The formula for the period [tex]\( P \)[/tex] of a cosine function [tex]\( h = A \cos (Bt) \)[/tex] is:
[tex]\[ P = \frac{2\pi}{B} \][/tex]
4. Substitute the value of [tex]\( B \)[/tex] into the period formula:
[tex]\[ P = \frac{2\pi}{\frac{\pi}{20}} \][/tex]
5. Simplify the expression:
[tex]\[ P = \frac{2\pi \cdot 20}{\pi} \][/tex]
6. Cancel out the [tex]\(\pi\)[/tex] in the numerator and denominator:
[tex]\[ P = 2 \cdot 20 \][/tex]
7. Multiply the remaining values:
[tex]\[ P = 40 \][/tex]
Therefore, the period of the given cosine function is 40 seconds. This means that it takes 40 seconds for the waterwheel to complete one full turn.
Thus, the correct answer is:
40 seconds
