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=acos(bt) to model the distance d, of the pendulum from the center (in inches) as a function of time t (in seconds). a=square

Question

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=acos(bt) to model the distance d, of the pendulum from the center (in inches) as a function of time t (in seconds). a=square

Answer 1

To model the distance

of the pendulum from the center as a function of time

, we need to determine the amplitude

and the frequency factor

for the cosine function \( d = a \cos(bt) \).Given:- The pendulum swings a horizontal distance of 12 inches from right to left and back in 1 second.The amplitude

is the maximum distance the pendulum reaches from the center. Since the pendulum swings 12 inches from the center to the right and then back, the amplitude

is 12 inches.So,

.Next, we need to determine the frequency factor

. The period of the cosine function is the time it takes for the pendulum to complete one full swing, which is 2 seconds (1 second to the right and 1 second to the left).The period

of a cosine function \( d = a \cos(bt) \) is given by

.Since the period

is 2 seconds, we have:

Solving for

:

Therefore, the cosine function that models the distance

of the pendulum from the center as a function of time

is:

So, the value of

is

.

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