1. **Problem statement:** We are given a sinusoidal function representing height over time and asked to create the equation for height, find the radius of the wheel, initial height, and total distance traveled. 2. **Equation for height over time:** The height oscillates between -1 and 11 meters, so amplitude $A = \frac{11 - (-1)}{2} = 6$ meters. The vertical shift (midline) is $D = \frac{11 + (-1)}{2} = 5$ meters. The period $T$ is $120$ seconds (from $170 - 50$). The angular frequency $k = \frac{2\pi}{T} = \frac{2\pi}{120} = \frac{\pi}{60}$. The phase shift is $50$ seconds. Thus, the height function is: $$H(t) = 6 \cos\left(\frac{\pi}{60}(t - 50)\right) + 5$$ 3. **Radius of the wheel:** The radius equals the amplitude, so radius $r = 6$ meters. 4. **Initial height:** Substitute $t=0$ into $H(t)$: $$H(0) = 6 \cos\left(\frac{\pi}{60}(0 - 50)\right) + 5 = 6 \cos\left(-\frac{5\pi}{6}\right) + 5 = 6 \times \left(-\frac{\sqrt{3}}{2}\right) + 5 = -5.196 + 5 = -0.20$$ meters (to 2 decimal places). 5. **Total distance traveled in 4 minutes (240 seconds):** Number of full cycles in 240 seconds: $$\frac{240}{120} = 2$$ Distance per cycle is twice the amplitude (max to min and back): Max height $= 11$, min height $= -1$, so max-min distance $= 12$ meters. Distance per cycle $= 2 \times 12 = 24$ meters. Total distance for 2 cycles: $$24 \times 2 = 48$$ meters. **Final answers:** - Height function: $H(t) = 6 \cos\left(\frac{\pi}{60}(t - 50)\right) + 5$ - Radius of wheel: $6$ meters - Initial height: $-0.20$ meters - Total distance traveled in 4 minutes: $48$ meters