To find the function that models the horizontal displacement of the pendulum as a function of time, we need to break down the problem into smaller steps.
1. Understand the swing time and distance:
- The given time for a one-way swing (from left to right) is [tex]\(1.5\)[/tex] seconds.
- The total horizontal distance covered is [tex]\(10\)[/tex] inches.
2. Determine the amplitude:
- The amplitude is half of the total distance covered. Therefore, the amplitude [tex]\(A\)[/tex] is:
[tex]\[
A = \frac{10}{2} = 5 \text{ inches}
\][/tex]
3. Calculate the period of the pendulum:
- A full period ([tex]\(T\)[/tex]) is the time it takes for one complete cycle (left to right and back to left). Since the given swing time is for one way, the period is twice the swing time:
[tex]\[
T = 1.5 \times 2 = 3 \text{ seconds}
\][/tex]
4. Find the angular frequency:
- The angular frequency [tex]\(\omega\)[/tex] is given by the formula:
[tex]\[
\omega = \frac{2\pi}{T}
\][/tex]
Substituting the value of [tex]\(T\)[/tex]:
[tex]\[
\omega = \frac{2\pi}{3} \text{ (rad/s)}
\][/tex]
5. Formulate the function for horizontal displacement:
- The displacement function generally has the form:
[tex]\[
y = A \cos (\omega x)
\][/tex]
where [tex]\(A\)[/tex] is the amplitude, [tex]\(\omega\)[/tex] is the angular frequency, and [tex]\(x\)[/tex] is the time in seconds.
6. Substitute the values:
- Using [tex]\(A = 5\)[/tex] inches and [tex]\(\omega = \frac{2\pi}{3}\)[/tex]:
[tex]\[
y = 5 \cos \left(\frac{2\pi}{3} x\right)
\][/tex]
This matches one of the given functions:
[tex]\[
y = 5 \cos \left(\frac{2 \pi}{3} x\right)
\][/tex]
In conclusion, the function that models the horizontal displacement of the pendulum as a function of time in seconds is:
[tex]\[
y = 5 \cos \left(\frac{2 \pi}{3} x\right)
\][/tex]
