Problemas
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
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Juanprofessionell · Tutor durante 6 años
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To model the distance \( d \) of the pendulum from the center as a function of time \( t \), we need to determine the amplitude \( a \) and the frequency factor \( b \) for the cosine function \( d = a \cos(bt) \).<br /><br />Given:<br />- The pendulum swings a horizontal distance of 12 inches from right to left and back in 1 second.<br /><br />The amplitude \( a \) 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 \( a \) is 12 inches.<br /><br />So, \( a = 12 \).<br /><br />Next, we need to determine the frequency factor \( b \). 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).<br /><br />The period \( T \) of a cosine function \( d = a \cos(bt) \) is given by \( T = \frac{2\pi}{b} \).<br /><br />Since the period \( T \) is 2 seconds, we have:<br />\[ 2 = \frac{2\pi}{b} \]<br /><br />Solving for \( b \):<br />\[ b = \frac{2\pi}{2} = \pi \]<br /><br />Therefore, the cosine function that models the distance \( d \) of the pendulum from the center as a function of time \( t \) is:<br />\[ d = 12 \cos(\pi t) \]<br /><br />So, the value of \( a \) is \( 12 \).
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