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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.R.37
Chapter 4, Problem 4.R.37

{Use of Tech } Minimizing sound intensity Two sound speakers are 100 m apart and one speaker is three times as loud as the other speaker. At what point on a line segment between the speakers is the sound intensity the weakest? (Hint: Sound intensity is directly proportional to the sound level and inversely proportional to the square of the distance from the sound source.)

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Define the sound intensity formula: Sound intensity (I) is proportional to the sound level (L) and inversely proportional to the square of the distance (d) from the source. Mathematically, this can be expressed as I = k * (L / d^2), where k is a proportionality constant.
Let the sound level of the quieter speaker be L1 and the louder speaker be L2 = 3L1. Place the quieter speaker at position x = 0 and the louder speaker at x = 100 on a coordinate line.
Consider a point P at a distance x from the quieter speaker and (100 - x) from the louder speaker. The sound intensity at point P due to the quieter speaker is I1 = k * (L1 / x^2) and due to the louder speaker is I2 = k * (3L1 / (100 - x)^2).
The total sound intensity at point P is the sum of the intensities from both speakers: I_total = I1 + I2 = k * (L1 / x^2) + k * (3L1 / (100 - x)^2).
To find the point where the sound intensity is weakest, differentiate I_total with respect to x, set the derivative equal to zero, and solve for x. This will give the critical points, which can be tested to find the minimum intensity.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sound Intensity

Sound intensity refers to the power per unit area carried by a sound wave. It is typically measured in watts per square meter (W/m²) and is directly related to the loudness of the sound. In this context, sound intensity is crucial for determining how the loudness from each speaker affects the overall sound intensity at different points along the line segment between them.
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Inverse Square Law

The inverse square law states that the intensity of a physical quantity (like sound) decreases with the square of the distance from the source. This means that if you double the distance from a sound source, the intensity becomes one-fourth. Understanding this principle is essential for calculating how the sound intensity from each speaker diminishes as you move away from it.
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Proportional Relationships

Proportional relationships describe how two quantities change in relation to each other. In this problem, sound intensity is directly proportional to the sound level of the speakers and inversely proportional to the square of the distance from each speaker. Recognizing these relationships allows for the formulation of equations to find the point where the combined sound intensity is minimized.
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