starting at its rightmost position, it takes 1 second for the pendulum of a grandfather clock to swing a…

starting at its rightmost position, it takes 1 second for the pendulum of a grandfather clock to swing a horizontal distance of 12 inches from right to left, and 1 second for the pendulum to swing back from left to right. write a cosine function, $d = a\\cos(bt)$, to model the distance, $d$, of the pendulum from the center (in inches) as a function of time $t$ (in seconds).\na = 6\ncomplete\nthe period is seconds.\nb = \ndone

starting at its rightmost position, it takes 1 second for the pendulum of a grandfather clock to swing a horizontal distance of 12 inches from right to left, and 1 second for the pendulum to swing back from left to right. write a cosine function, $d = a\\cos(bt)$, to model the distance, $d$, of the pendulum from the center (in inches) as a function of time $t$ (in seconds).\na = 6\ncomplete\nthe period is seconds.\nb = \ndone

Answer

Explanation:

Step1: Determine the amplitude

The pendulum swings 12 inches from right - to - left. The amplitude $a$ is half of the total horizontal distance it swings. So $a=\frac{12}{2}=6$.

Step2: Find the period

It takes 1 second to swing from right to left and 1 second to swing back from left to right. So the period $T$ (the time for one full back - and - forth motion) is $T = 2$ seconds.

Step3: Calculate the value of $b$

The formula for the period of a cosine function $y = a\cos(bt)$ is $T=\frac{2\pi}{b}$. Since $T = 2$, we can solve for $b$: $2=\frac{2\pi}{b}$, then $b=\pi$.

Answer:

The period is 2 seconds. $b=\pi$