1. **State the problem:** We are given graphs of the x and y coordinates of the tip of the hour hand of a clock as functions of time $t$ in hours. We need to find initial coordinates, initial time, and equations for $x(t)$ and $y(t)$. 2. **Initial coordinates:** From the graph description, the x-coordinate at $t=0$ is $-4$ and the y-coordinate at $t=0$ is $5$. So the initial coordinates are $(-4,5)$. 3. **Initial time:** The hour hand points at the initial position $(-4,5)$. Since the clock center is at $(0,0)$, the angle $ heta$ of the hour hand at $t=0$ satisfies: $$\cos(\theta) = \frac{x}{r} = \frac{-4}{5}, \quad \sin(\theta) = \frac{y}{r} = \frac{5}{5} = 1$$ where $r=5$ is the amplitude (length of hour hand). Since $\sin(\theta)=1$ and $\cos(\theta)=-0.8$, $\theta = 90^\circ + \alpha$ where $\alpha$ is small. This corresponds to 10 o'clock (since hour hand moves $30^\circ$ per hour, $90^\circ + 30^\circ = 120^\circ$ is 4 hours from 12, but here the angle is about $128.7^\circ$ which corresponds to about 10 o'clock). So initial time is 10:00. 4. **Equation for $x(t)$:** The x-coordinate is a cosine wave with amplitude $a=5$ (length of hour hand) but the graph shows amplitude about 4, so $a=5$ is correct from the y amplitude, but the graph says amplitude about 4 for x, so $a=5$ for y and $a=4$ for x. The period of the hour hand is 12 hours, so angular frequency $b = \frac{360^\circ}{12} = 30^\circ$ per hour. The x-coordinate starts at $-4$ at $t=0$, so: $$x(t) = 4 \cos(30^\circ t + \phi)$$ Since $x(0) = 4 \cos(\phi) = -4$, then $\cos(\phi) = -1$, so $\phi = 180^\circ$. Thus: $$x(t) = 4 \cos(30^\circ t + 180^\circ)$$ 5. **Equation for $y(t)$:** The y-coordinate is a sine wave with amplitude 5, starting at $y(0) = 5$, so: $$y(t) = 5 \sin(30^\circ t + \psi)$$ Since $y(0) = 5 \sin(\psi) = 5$, then $\sin(\psi) = 1$, so $\psi = 90^\circ$. Thus: $$y(t) = 5 \sin(30^\circ t + 90^\circ)$$ **Final answers:** - Initial coordinates: $(-4,5)$ - Initial time: 10:00 - $x(t) = 4 \cos(30^\circ t + 180^\circ)$ - $y(t) = 5 \sin(30^\circ t + 90^\circ)$