1. **State the problem:** We are given the function $c(t) = 28 + 4 \sin\left(\frac{\pi}{2} t\right)$ and we want to understand its behavior. 2. **Formula and explanation:** This is a sinusoidal function of the form $c(t) = A + B \sin(\omega t)$ where: - $A = 28$ is the vertical shift (midline). - $B = 4$ is the amplitude (maximum deviation from the midline). - $\omega = \frac{\pi}{2}$ is the angular frequency. 3. **Important rules:** - The sine function oscillates between $-1$ and $1$. - Therefore, $c(t)$ oscillates between $28 - 4 = 24$ and $28 + 4 = 32$. - The period $T$ of the sine function is given by $T = \frac{2\pi}{\omega} = \frac{2\pi}{\frac{\pi}{2}} = 4$. 4. **Intermediate work:** - Calculate the period: $$T = \frac{2\pi}{\frac{\pi}{2}} = 2\pi \times \frac{2}{\pi} = 4$$ - The function completes one full cycle every 4 units of $t$. 5. **Summary:** - The function $c(t)$ oscillates between 24 and 32. - It has a midline at 28. - It completes one full oscillation every 4 units of $t$. **Final answer:** The function $c(t) = 28 + 4 \sin\left(\frac{\pi}{2} t\right)$ oscillates between 24 and 32 with period 4 and midline 28.