Let's solve the problem step by step: ## **Step 1: Understanding the Problem** - **HVAC alone:** 55 dB - **Fan alone:** 60 dB - **Both running:** What is the combined sound pressure level (dB)? Sound levels in dB **cannot be added arithmetically** because dB is a logarithmic unit. ## **Step 2: Formula to Combine Sound Levels** For two sound sources: \[ L_{\text{total}} = 10 \log_{10} \left(10^{L_1/10} + 10^{L_2/10}\right) \] Where: - \( L_1 \) = Sound level of source 1 (HVAC) = **55 dB** - \( L_2 \) = Sound level of source 2 (Fan) = **60 dB** ## **Step 3: Plug in the Numbers** Calculate the power equivalents: \[ 10^{L_1/10} = 10^{55/10} = 10^{5.5} \] \[ 10^{L_2/10} = 10^{60/10} = 10^{6} \] So: \[ L_{\text{total}} = 10 \log_{10} \left(10^{5.5} + 10^{6}\right) \] Let’s compute \( 10^{5.5} \) and \( 10^{6} \): - \( 10^{5.5} = 316227.766 \) - \( 10^{6} = 1000000 \) Add them: \[ 316227.766 + 1000000 = 1316227.766 \] ## **Step 4: Take the Logarithm** \[ L_{\text{total}} = 10 \log_{10}(1316227.766) \] \[ \log_{10}(1316227.766) \approx 6.119 \] \[ L_{\text{total}} = 10 \times 6.119 = 61.19\, \text{dB} \] ## **Step 5: Final Answer** \[ \boxed{61.2\, \text{dB}} \] ### **Explanation:** - Decibels are logarithmic, not linear. - You convert each dB level to its power equivalent, sum them, then convert back to dB. - The resulting sound level is **not** the arithmetic sum, but a little above the higher of the two. --- ### **Final Answer:** > **The combined sound pressure level when both the HVAC and the fan are running is approximately \(\boxed{61.2\, \text{dB}}\).**

**Introduction** In acoustics, sound levels are commonly expressed in decibels (dB), a logarithmic unit that quantifies the intensity of sound relative to a reference level. When multiple sound sources operate simultaneously, their combined effect must be calculated carefully because decibels cannot be added arithmetically; instead, they require conversion to their linear power equivalents, summation, and then conversion back to decibels. Understanding how to combine sound levels is fundamental in sound engineering, noise control, and acoustic analysis, especially when evaluating the cumulative noise produced by multiple sources such as HVAC systems and fans. The problem involves finding the resultant sound pressure level when both an HVAC and a fan, each producing known sound levels independently, operate together. **Explanation:** This introduction emphasizes the importance of the logarithmic nature of decibel measurements and outlines the necessity of converting dB levels to linear power units for accurate combination. It provides the foundational understanding that the problem’s core involves the principles of acoustic summation, which is essential for subsequent calculations and comprehension. --- **Presentation of Relevant Formulas Required To Solve The Question** 1. **Conversion from decibels to power ratio:** \[ L = 10 \log_{10} \left(\frac{P}{P_0}\right) \quad \Rightarrow \quad P = P_0 \times 10^{L/10} \] Where: - \(L\) is the sound level in decibels (dB), - \(P\) is the sound power, - \(P_0\) is the reference power (commonly \(10^{-12}\) W, but the absolute value cancels out when ratios are used). 2. **Combining two sound sources:** \[ L_{\text{total}} = 10 \log_{10} \left(10^{L_1/10} + 10^{L_2/10}\right) \] Where: - \(L_1, L_2\) are the individual sound levels in decibels. **Explanation:** The first formula translates a decibel level into a linear power ratio, which enables the addition of contributions from multiple sources. The second formula is derived from the fact that sound pressure levels relate logarithmically to power; summing the powers of individual sources and converting back to decibels yields the combined level. These formulas are fundamental in acoustic calculations involving multiple sources. --- **A Detailed Step-by-Step Solution** **Step 1: Convert individual sound levels from decibels to their linear power equivalents** - For the HVAC system: \[ P_{HVAC} = 10^{55/10} = 10^{5.5} \approx 316227.766 \] - For the Fan: \[ P_{Fan} = 10^{60/10} = 10^{6} = 1,000,000 \] **Explanation:** Converting the decibel levels to their linear power equivalents allows for straightforward addition, as decibel addition is nonlinear. This step establishes the basis for combining the individual sound contributions accurately. --- **Step 2: Sum the linear powers** \[ P_{total} = P_{HVAC} + P_{Fan} = 316227.766 + 1,000,000 = 1,316,227.766 \] **Explanation:** Adding the power equivalents reflects the total acoustic energy contributed by both sources when operating simultaneously. This sum represents the combined effect in the linear domain, where simple addition applies. --- **Step 3: Convert the total power back to decibels** \[ L_{total} = 10 \log_{10} (P_{total}) = 10 \log_{10} (1,316,227.766) \] Calculating the logarithm: \[ \log_{10} (1,316,227.766) \approx 6.119 \] Multiplying by 10: \[ L_{total} = 10 \times 6.119 = 61.19\, \text{dB} \] **Explanation:** Reverting to decibels provides the combined sound level in familiar units. The value obtained reflects the logarithmic sum of the individual sound sources, ensuring an accurate representation of the total noise level. --- **Conclusion** The combined sound pressure level when both the HVAC system and the fan are operating simultaneously is approximately **61.2 dB**. This value is slightly higher than the individual levels due to the superposition of their acoustic energies, demonstrating the importance of logarithmic addition in acoustic calculations. **Final Answer:** > **The total sound pressure level when both the HVAC and the fan run together is approximately \(\boxed{61.2\, \text{dB}}\).**