1. **Problem 3:** How many times louder is a live rock concert at 108 dB compared to a vacuum cleaner at 70 dB? The formula to compare loudness levels in decibels is: $$L = 10 \log_{10}\left(\frac{I_1}{I_2}\right)$$ where $L$ is the difference in decibels, $I_1$ and $I_2$ are the intensities. Rearranged to find the ratio of intensities: $$\frac{I_1}{I_2} = 10^{\frac{L}{10}}$$ Calculate: $$\frac{I_1}{I_2} = 10^{\frac{108 - 70}{10}} = 10^{3.8}$$ Using a calculator: $$10^{3.8} \approx 6310.\n$$ So, the live rock concert is approximately 6310 times louder than the vacuum cleaner. 2. **Problem 4:** Given the function $$h(t) = 10 \sin\left(\frac{\pi}{30}(t - 5)\right) + 12,$$ where $h$ is height in meters and $t$ is time in seconds. (a) **Period and Amplitude:** - Amplitude $A$ is the coefficient before sine: $A = 10$ meters. - Period $T$ is given by: $$T = \frac{2\pi}{b}$$ where $b = \frac{\pi}{30}$. Calculate period: $$T = \frac{2\pi}{\frac{\pi}{30}} = 2\pi \times \frac{30}{\pi} = 60 \text{ seconds}.$$ **Meaning:** - Amplitude 10 means the Ferris wheel moves 10 meters above and below the center height. - Period 60 seconds means it takes 60 seconds for one full rotation. (b) **Height at $t=35$ seconds:** Calculate inside sine: $$\theta = \frac{\pi}{30}(35 - 5) = \frac{\pi}{30} \times 30 = \pi.$$ Evaluate: $$h(35) = 10 \sin(\pi) + 12 = 10 \times 0 + 12 = 12 \text{ meters}.$$ (c) **First time height is 20 m:** Set $h(t) = 20$: $$20 = 10 \sin\left(\frac{\pi}{30}(t - 5)\right) + 12$$ Subtract 12: $$8 = 10 \sin\left(\frac{\pi}{30}(t - 5)\right)$$ Divide both sides by 10: $$\frac{8}{10} = \sin\left(\frac{\pi}{30}(t - 5)\right)$$ $$0.8 = \sin\left(\frac{\pi}{30}(t - 5)\right)$$ Take inverse sine: $$\frac{\pi}{30}(t - 5) = \sin^{-1}(0.8) \approx 0.9273$$ Solve for $t$: $$t - 5 = \frac{30}{\pi} \times 0.9273 \approx 8.85$$ $$t = 8.85 + 5 = 13.85 \text{ seconds}.$$ (d) **Next time height is 20 m:** Since sine is periodic, the next solution in $[0, 2\pi]$ is: $$\frac{\pi}{30}(t - 5) = \pi - 0.9273 = 2.2143$$ Solve for $t$: $$t - 5 = \frac{30}{\pi} \times 2.2143 \approx 21.15$$ $$t = 21.15 + 5 = 26.15 \text{ seconds}.$$ **Final answers:** - Problem 3: The concert is approximately 6310 times louder. - Problem 4(a): Amplitude = 10 m, Period = 60 s. - Problem 4(b): Height at 35 s = 12 m. - Problem 4(c): First time height = 20 m at $t \approx 13.85$ s. - Problem 4(d): Next time height = 20 m at $t \approx 26.15$ s.