Problem 46 - Technical Mathematics with Calculus

Question

Applications. For a certain device, \(P_{1}=4.0 \mathrm{W}\) and \(P_{2}=280 \mathrm{W} .\) Find the change in decibel level.

Step-by-Step Solution

Answer

\( \Delta L \approx 18.5 \) decibels

Step 1: Understanding the Problem

The exercise asks us to find the change in decibel level for a certain device when its power level changes from 4.0 W to 280 W. The decibel level can be found using the formula: \( \Delta L = 10 \cdot \log_{10}\left(\frac{P_2}{P_1}\right) \), where \( \Delta L \) is the change in decibel level, \( P_1 \) is the initial power level, and \( P_2 \) is the final power level.

Step 2: Substituting the Given Values

Substitute the given values for \( P_1 \) and \( P_2 \) into the decibel change formula: \( \Delta L = 10 \cdot \log_{10}\left(\frac{280}{4.0}\right) \).

Step 3: Solving for the Change in Decibel Level

Calculate the fraction inside the logarithm and then the logarithm itself to find the change in decibel level: \( \Delta L = 10 \cdot \log_{10}(70) \approx 10 \cdot 1.8451 = 18.451 \Rightarrow \Delta L \approx 18.5 \) decibels.

Key Concepts

Logarithmic Calculations

When encountering problems involving sound intensity and decibels, understanding logarithmic calculations is essential. Logarithms are a way to express exponents or powers in terms of calculations that can more easily compare very large or very small numbers. The base-10 logarithm, denoted as \( \log_{10} \), is particularly common in scientific calculations, including those involving decibels. In the context of decibel calculations, the logarithm helps us describe the relative change in power or intensity.

For example, if we have two power levels, \( P_{1} \) and \( P_{2} \) and want to find the relative change in terms of decibels, we use the logarithmic formula \( \Delta L = 10 \cdot \log_{10}\left(\frac{P_2}{P_1}\right) \). This formula hinges on our understanding of how to handle logarithms, calculate the ratio of powers, and finally multiply the result by 10 to convert it to decibels. A key property of logarithms used here is that the log of a quotient is the difference of the logs, which translates into the decibel difference in this scenario.

Power Level in Watts

The power level of a device, commonly measured in watts (W), plays a crucial role in understanding energy consumption and sound intensity. A watt is a derived unit of power in the International System of Units (SI) that measures the rate at which energy is used. In the case of sound, the power level corresponds to the acoustic power radiated by a sound source.

Understanding the power level in watts is key when discussing changes in decibel levels because the decibel scale is a logarithmic measure of the ratio between two power levels. Consequently, even a small change in power measured in watts can result in a significant change in decibels. This sensitivity to change is why we often use decibels to express audio volume levels, as it correlates better with human perception of loudness compared to a linear scale.

Decibel Formula Application

The application of the decibel formula, \( \Delta L = 10 \cdot \log_{10}\left(\frac{P_2}{P_1}\right) \), allows us to quantify the relative change in power level in a way that is more attuned to human auditory perception. The decibel (dB) is a logarithmic unit used to express the ratio of two values of a physical quantity, often power or intensity.

In our textbook exercise, the decibel level change correlates to how much louder or quieter a device becomes when its power level changes from 4.0 W to 280 W. We calculate this by finding the ratio of the two power levels and then applying the logarithmic function. The result, multiplied by 10, gives us a change in decibels, a unit that is more meaningful to our ears than a simple ratio of power levels. This practical application demonstrates how logarithmic calculations are not just abstract mathematical concepts but are used in real-world measurements and can affect everyday experiences, like listening to music or measuring the noise level in a busy street.