\begin{tabular}{l} An object moves in simple harmonic motion described by the equation \( d=-2 \sin \frac{2 \pi}{7} t \) 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. \\ 2 in. \\ (Type an integer or a fraction.) \\ b. Find the frequency. \\ \( \square \) cycles per second \\ (Type an exact answer, using \( \pi \) as needed. Use integers or fractions for any numbers in the expression.) \\ \hline\end{tabular}

**Step 1. Identify the amplitude (maximum displacement).** The displacement function is \[ d = -2 \sin\left(\frac{2\pi}{7}t\right). \] The amplitude is the absolute value of the coefficient of the sine function. Thus, the maximum displacement is \[ \left| -2 \right| = 2 \text{ inches}. \] **Step 2. Determine the angular frequency and frequency.** From the equation, the angular frequency is \[ \omega = \frac{2\pi}{7} \text{ radians per second}. \] The frequency is the number of cycles per second and is given by \[ f = \frac{\omega}{2\pi}. \] Substitute the value of \(\omega\): \[ f = \frac{\frac{2\pi}{7}}{2\pi} = \frac{1}{7} \text{ cycles per second}. \] **Step 3. Calculate the time required for one cycle (the period).** The period \(T\) is the reciprocal of the frequency: \[ T = \frac{1}{f}. \] Substitute the frequency \(f = \frac{1}{7}\): \[ T = \frac{1}{\frac{1}{7}} = 7 \text{ seconds}. \] **Final Answers:** - **a. Maximum displacement:** \(2\) inches - **b. Frequency:** \(\frac{1}{7}\) cycles per second - **Time required for one cycle:** \(7\) seconds