The noise rating of an air conditioner is 50 decibels, and the noise rating of a washing machine is 65 decibels. Determine how many time greater the washing machine's intensity of sound is. Use the formula \( \beta=10 \log \frac{I}{I_{0}} \), where \( \beta \) is the sound level in decibels, \( l \) is the intensity of sound you are measuring, and \( I_{0} \) is the smallest sound intensity that can be heard by the human ear (roughly equal to \( 1 \times 10^{-12} \) watts \( / \mathrm{m}^{2} \) ). The washing machine's intensity of sound is

To calculate how many times greater the washing machine's intensity of sound is compared to the air conditioner, we'll start by using the given formula. For the air conditioner: \[ 50 = 10 \log \frac{I_{AC}}{1 \times 10^{-12}} \] Rearranging gives: \[ \log \frac{I_{AC}}{1 \times 10^{-12}} = 5 \implies \frac{I_{AC}}{1 \times 10^{-12}} = 10^5 \implies I_{AC} = 10^5 \times 1 \times 10^{-12} = 1 \times 10^{-7} \text{ watts/m}^2 \] For the washing machine: \[ 65 = 10 \log \frac{I_{WM}}{1 \times 10^{-12}} \] Rearranging gives: \[ \log \frac{I_{WM}}{1 \times 10^{-12}} = 6.5 \implies \frac{I_{WM}}{1 \times 10^{-12}} = 10^{6.5} \implies I_{WM} = 10^{6.5} \times 1 \times 10^{-12} \approx 3.16 \times 10^{-6} \text{ watts/m}^2 \] Now to find how many times greater the intensity of the washing machine is compared to the air conditioner: \[ \frac{I_{WM}}{I_{AC}} = \frac{3.16 \times 10^{-6}}{1 \times 10^{-7}} = 31.6 \] So, the washing machine's intensity of sound is approximately 31.6 times greater than that of the air conditioner. The washing machine's intensity of sound is around 31.6 times greater than the air conditioner's! The fascinating evolution of home appliances has truly transformed our way of living. Did you know that early washing machines were hand-cranked, often resulting in more backache than clean clothes? Innovations like automatic cycles and energy-efficient models now help us save time and money, not to mention quieting down our homes, even if the washing machine is still the loudest contestant in the appliance showdown! If you're looking to minimize the noise from your appliances, consider placing them on rubber mats; this technique minimizes vibrations and can significantly lower sound levels. Also, check if you're running appliances when you really need them, as running multiple noisy ones at the same time can lead to a cacophonous household!