1. **State the problem:** We need to find the value of $d$ in the equation $y = a \sin(bx + c) + d$ that models the height $y$ of a student on a ferris wheel after $x$ seconds. 2. **Given information:** - Diameter of the ferris wheel = 12 meters - The student starts at the bottom at $t=0$ at a height of 3 meters 3. **Understand the parameters:** - The amplitude $a$ is half the diameter, so $a = \frac{12}{2} = 6$ meters. - The vertical shift $d$ is the height of the center of the wheel above the ground. 4. **Key fact:** The lowest point on the wheel is at height $d - a$. 5. Since the student starts at the bottom at $t=0$, the height at $x=0$ is $y = d - a = 3$ meters. 6. Substitute $a=6$: $$d - 6 = 3$$ 7. Solve for $d$: $$d = 3 + 6 = 9$$ **Final answer:** $$d = 9$$ meters This means the center of the ferris wheel is 9 meters above the ground.