Question 2 a) Two audio systems \( \mathbf{A} \) and \( \mathbf{B} \) produce a sound intensity levels \( \beta_{A}=90.0 \) and \( \beta_{\mathrm{B}}=93.0 \mathrm{~dB} \). Determine the ratio of intensity \( \mathbf{A} \) to intensity \( \mathbf{B} \).

To determine the ratio of intensity \( A \) to intensity \( B \), we can use the formula for sound intensity levels: \[ \beta = 10 \log_{10} \left( \frac{I}{I_0} \right) \] where: - \( \beta \) is the sound intensity level in decibels (dB), - \( I \) is the intensity of the sound, - \( I_0 \) is the reference intensity (usually \( 10^{-12} \) W/m²). Given that the sound intensity levels are \( \beta_A = 90.0 \) dB and \( \beta_B = 93.0 \) dB, we can calculate the ratio of intensity \( A \) to intensity \( B \) using the formula: \[ \frac{I_A}{I_B} = \frac{10^{(\beta_A - 90)/10}}{10^{(\beta_B - 90)/10}} \] Substitute the given values into the formula to find the ratio of intensity \( A \) to intensity \( B \). Calculate the value by following steps: - step0: Calculate: \(\frac{10^{\frac{\left(90-90\right)}{10}}}{10^{\frac{\left(93-90\right)}{10}}}\) - step1: Remove the parentheses: \(\frac{10^{\frac{90-90}{10}}}{10^{\frac{93-90}{10}}}\) - step2: Multiply by \(a^{-n}:\) \(10^{\frac{90-90}{10}}\times 10^{-\frac{93-90}{10}}\) - step3: Subtract the terms: \(10^{\frac{0}{10}}\times 10^{-\frac{93-90}{10}}\) - step4: Subtract the numbers: \(10^{\frac{0}{10}}\times 10^{-\frac{3}{10}}\) - step5: Divide the terms: \(10^{0}\times 10^{-\frac{3}{10}}\) - step6: Evaluate the power: \(1\times 10^{-\frac{3}{10}}\) - step7: Multiply: \(10^{-\frac{3}{10}}\) - step8: Express with a positive exponent: \(\frac{1}{10^{\frac{3}{10}}}\) - step9: Simplify: \(\frac{1}{\sqrt[10]{10^{3}}}\) - step10: Rewrite the expression: \(\frac{1}{\sqrt[10]{1000}}\) - step11: Multiply by the Conjugate: \(\frac{\sqrt[10]{1000^{9}}}{\sqrt[10]{1000}\times \sqrt[10]{1000^{9}}}\) - step12: Simplify: \(\frac{10^{2}\sqrt[10]{10^{7}}}{\sqrt[10]{1000}\times \sqrt[10]{1000^{9}}}\) - step13: Multiply the numbers: \(\frac{10^{2}\sqrt[10]{10^{7}}}{10^{3}}\) - step14: Reduce the fraction: \(\frac{\sqrt[10]{10^{7}}}{10}\) The ratio of intensity \( A \) to intensity \( B \) is approximately 0.501187.