[tex]\text{variation in pressure is }4635.9549\text{ pounds per square inch}[/tex]Explanation:
The formula relating the variation in pressure with loudness of sound:
[tex]\begin{gathered} L=20log_{10}(121.3P) \\ L\text{ = loundness of sound in decimals} \\ P\text{ = variation of pressure in pounds per square inch} \end{gathered}[/tex]
when L = 115 decibels
P = ?
we will susbtitute 115 for L in the formula above:
[tex]\begin{gathered} 115\text{ = }20log_{10}\text{ (121.3P)} \\ \text{divide both sides by 20:} \\ \frac{115}{20}\text{ = }\frac{20log_{10}\text{ (121.3P)}}{20} \\ 5.75\text{ = }log_{10}\text{ (121.3P)} \end{gathered}[/tex][tex]\begin{gathered} In\text{ }logarithm\colon a=log_bC \\ b^a\text{ = C} \\ \\ \text{Applying the principle here:} \\ 5.75\text{ = }log_{10}\text{ (121.3P)} \\ 10^{5.75}=\text{ 121.3P} \\ 562341.325\text{ = 121.3P} \\ \text{divide both sides by 121.3:} \\ \frac{562341.325}{121.3}\text{ = }\frac{\text{ 121.3P}}{121.3} \end{gathered}[/tex][tex]\begin{gathered} \frac{562341.325}{121.3}\text{ = }P \\ \text{P = }4635.95486 \\ \\ To\text{ the 4th decimal, variation in pressure is }4635.9549\text{ pounds per square inch} \end{gathered}[/tex]
