1. **Problem 1: Find the period of the pendulum ride given by** $$P(t) = -10 \sin\left(\frac{\pi}{3} t\right)$$ 2. The general form of a sine function is $$y = A \sin(Bt)$$ where the period $$T = \frac{2\pi}{B}$$. 3. Here, $$B = \frac{\pi}{3}$$, so the period is: $$ T = \frac{2\pi}{\frac{\pi}{3}} = 2\pi \times \frac{3}{\pi} = 6 $$ 4. This means the pendulum completes one full cycle in 6 seconds. --- 1. **Problem 2: Find the maximum brightness of the star given by** $$y = 2 \sin\left(\frac{\pi}{4} x\right) + 6$$ 2. The sine function oscillates between -1 and 1, so: $$ \sin\left(\frac{\pi}{4} x\right) \in [-1,1] $$ 3. Multiply by 2: $$ 2 \sin\left(\frac{\pi}{4} x\right) \in [-2,2] $$ 4. Add 6 to shift the range: $$ y \in [6 - 2, 6 + 2] = [4, 8] $$ 5. Therefore, the maximum brightness is 8.