1. **State the problem:** We need to find an equation for the sensor's height $y$ in centimeters as a function of time $t$ in seconds, given the sinusoidal motion with key points and period. 2. **Identify key values:** - Maximum height $y_{max} = 60$ cm - Minimum height $y_{min} = 38$ cm - Midline (average height) $M = \frac{60 + 38}{2} = 49$ cm - Amplitude $A = \frac{60 - 38}{2} = 11$ cm - Period $T = 3$ seconds (from $t=0.75$ to $t=3.75$) 3. **Write the general sinusoidal function:** $$y = M + A \sin\left(\frac{2\pi}{T}(t - \phi)\right)$$ where $\phi$ is the phase shift. 4. **Determine phase shift $\phi$:** The maximum occurs at $t=0.75$, and sine reaches maximum at $\frac{\pi}{2}$. So, $$\frac{2\pi}{T}(0.75 - \phi) = \frac{\pi}{2}$$ Substitute $T=3$: $$\frac{2\pi}{3}(0.75 - \phi) = \frac{\pi}{2}$$ Divide both sides by $\pi$: $$\frac{2}{3}(0.75 - \phi) = \frac{1}{2}$$ Multiply both sides by 3: $$2(0.75 - \phi) = \frac{3}{2}$$ $$1.5 - 2\phi = 1.5$$ Subtract 1.5: $$-2\phi = 0$$ Divide by -2: $$\phi = 0$$ 5. **Final equation:** $$y = 49 + 11 \sin\left(\frac{2\pi}{3} t \right)$$ This equation models the sensor's height over time. **Answer:** $$\boxed{y = 49 + 11 \sin\left(\frac{2\pi}{3} t \right)}$$