1. **State the problem:** We have a periodic function $h(t)$ modeling the distance between point B on a rotating boat motor blade and the water line. The motor blades rotate 50 times per second, point B is 8 inches from the center, and the center is 18 inches below the water line. 2. **Analyze the periodic motion:** - The rotation frequency is 50 rotations per second, so the period is $T = \frac{1}{50} = 0.02$ seconds. - The distance $h(t)$ varies sinusoidally with amplitude 8 inches (distance from center to point B). - The midline (average distance) is the center's distance from the water line, 18 inches. 3. **Coordinates of points on the graph:** - The midline is at $h = 18$. - The amplitude is 8, so the maximum height (peak) is $18 + 8 = 26$ inches. - The minimum height (trough) is $18 - 8 = 10$ inches. 4. **Identify points:** - $F$: peak of first cycle at $t=0$, so $(0, 26)$. - $G$: midline crossing descending, occurs at $t = \frac{T}{4} = 0.005$, so $(0.005, 18)$. - $J$: trough at $t = \frac{T}{2} = 0.01$, so $(0.01, 10)$. - $K$: midline crossing ascending at $t = \frac{3T}{4} = 0.015$, so $(0.015, 18)$. - $P$: peak of second cycle at $t = T = 0.02$, so $(0.02, 26)$. 5. **Answer part (B)(i):** On the interval $(t_1, t_2) = (0.005, 0.01)$, $h(t)$ goes from midline descending to trough. - $h$ is positive (distance above 0) and decreasing. - Correct choice: b. $h$ is positive and decreasing. 6. **Answer part (B)(ii):** The rate of change of $h$ is the slope of the curve. - Between $t_1$ and $t_2$, $h$ is decreasing but the slope is becoming less steep as it approaches the trough. - So, the rate of change is negative but increasing (becoming less negative). Final answers: - $F = (0, 26)$ - $G = (0.005, 18)$ - $J = (0.01, 10)$ - $K = (0.015, 18)$ - $P = (0.02, 26)$ - (B)(i) Answer: b. $h$ is positive and decreasing. - (B)(ii) The rate of change of $h$ is negative but increasing over $(t_1, t_2)$.