1. **Problem statement:** We have hot chocolate amounts measured hourly from 9am to 5pm: 300, 100, 300, 500, 300, 100, 300, 500, 300 mL. We want to find the amplitude, equation of axis, period, wave number $k$, and the coefficients $d_{\cos}$ and $d_{\sin}$ of the sinusoidal function modeling this data. 2. **Step 1: Identify amplitude and axis equation.** - The maximum value is 500 mL and the minimum is 100 mL. - Amplitude $A = \frac{\text{max} - \text{min}}{2} = \frac{500 - 100}{2} = 200$ mL. - Equation of axis (midline) $= \frac{\text{max} + \text{min}}{2} = \frac{500 + 100}{2} = 300$ mL. 3. **Step 2: Determine the period.** - The pattern repeats every 4 hours (e.g., 9am to 1pm: 300,100,300,500, then repeats). - So, period $T = 4$ hours. 4. **Step 3: Calculate wave number $k$.** - Formula: $k = \frac{2\pi}{T} = \frac{2\pi}{4} = \frac{\pi}{2}$. 5. **Step 4: Form the general sinusoidal equation.** - The function can be written as: $$y = A \sin(k(t - d)) + \text{axis}$$ or $$y = d_{\sin} \sin(kt) + d_{\cos} \cos(kt) + \text{axis}$$ 6. **Step 5: Find $d_{\sin}$ and $d_{\cos}$.** - Using the data at $t=0$ (9am), $y=300$: $$300 = d_{\sin} \sin(0) + d_{\cos} \cos(0) + 300 \Rightarrow 300 = 0 + d_{\cos} \times 1 + 300 \Rightarrow d_{\cos} = 0$$ - Using $t=1$ (10am), $y=100$: $$100 = d_{\sin} \sin\left(\frac{\pi}{2} \times 1\right) + 0 + 300 \Rightarrow 100 = d_{\sin} \times 1 + 300 \Rightarrow d_{\sin} = 100 - 300 = -200$$ 7. **Step 6: Final equation:** $$y = -200 \sin\left(\frac{\pi}{2} t\right) + 300$$ --- **Answer to part (a):** - Amplitude $= 200$ - Equation of axis $= y = 300$ - Period $= 4$ hours - Wave number $k = \frac{\pi}{2}$ - $d_{\cos} = 0$ - $d_{\sin} = -200$ --- **Part (b):** Find hot chocolate amount at 11:45am (i.e., $t=2.75$ hours after 9am). 1. Use the equation: $$y = -200 \sin\left(\frac{\pi}{2} \times 2.75\right) + 300$$ 2. Calculate inside the sine: $$\frac{\pi}{2} \times 2.75 = \frac{\pi}{2} \times \frac{11}{4} = \frac{11\pi}{8}$$ 3. Evaluate $\sin\left(\frac{11\pi}{8}\right)$: - $\sin\left(\frac{11\pi}{8}\right) = -\sin\left(\frac{3\pi}{8}\right) \approx -0.924$ (since $\sin(\pi + x) = -\sin x$) 4. Substitute: $$y = -200 \times (-0.924) + 300 = 184.8 + 300 = 484.8$$ 5. So, expected hot chocolate amount at 11:45am is approximately 485 mL. --- **Summary:** - The sinusoidal model is $y = -200 \sin\left(\frac{\pi}{2} t\right) + 300$ - At 11:45am ($t=2.75$), expected amount is about 485 mL.