An object moves in simple harmonic motion described by the equation \( d=\frac{1}{6} \sin 6 t \) where \( t \) is measured in seconds and \( d \) in inches. Find the maximum displacement, the frequency, and the time required for one cycle. a. Find the maximum displacement. \( \square \) in. (Type an integer or a fraction.)

The maximum displacement in simple harmonic motion occurs at the amplitude of the sine function. In the equation \( d=\frac{1}{6} \sin 6 t \), the coefficient of the sine function, \( \frac{1}{6} \), represents the maximum displacement. Therefore, the maximum displacement is \( \frac{1}{6} \) inches. To find the frequency of the motion, we use the angular frequency present in the equation. The \( 6 \) before \( t \) indicates the angular frequency \( \omega \). The frequency \( f \) can be found using the relationship \( f = \frac{\omega}{2\pi} \), so \( f = \frac{6}{2\pi} = \frac{3}{\pi} \) Hz. This tells us how many cycles happen in one second! The time required for one complete cycle, or the period \( T \), is the reciprocal of the frequency: \( T = \frac{1}{f} = \frac{2\pi}{6} = \frac{\pi}{3} \) seconds.