Question

Given a sound source (e.g. violin in an orchestra), how many violins do we need so that we perceive their sound as twice as loud? The answer is 10 . To make it 4 times as loud (to our senses), we need 100 violins! This relationship is modelled by the function y=10ˡᵒᵍ₂⁽ˣ⁾ , where y is the number of violins and x is how loud many times louder we perceive it as compared to 1 violin. (A) Interpret what y⁻¹ describes about violins and loudness factor. (B) Interpret in words what d(y⁻¹)/dx describes about violins and loudness factor. (C) Write a mathematical equation that explains what it means for y⁻¹ to be the inverse function of y. (D) Use the Chain "Rule" to take the derivative of the equation from part (C) with respect to x. (E) Solve for d(y⁻¹)/dx . Explain your answers in detail.

Answer 1

(A) y⁻¹ describes the reciprocal relationship between the number of violins and the perceived loudness factor.

B. (B) The interpretation of d(y⁻¹)/dx is the rate of change of the reciprocal of the number of violins concerning the change in the perceived loudness factor.

Step-by-step explanation:

The reciprocal function y⁻¹ represents the inverse relationship between the number of violins and the perceived loudness factor. As the number of violins increases (y), the perceived loudness factor decreases inversely (y⁻¹). In essence, y⁻¹ quantifies how many times louder a single violin would sound in comparison to the total number of violins.

When evaluating d(y⁻¹)/dx, it represents the rate of change of the reciprocal of violins with respect to the change in the perceived loudness factor. This derivative showcases how rapidly the perceived loudness factor changes concerning the number of violins employed. A higher d(y⁻¹)/dx value indicates that the change in the perceived loudness factor is more sensitive to adjustments in the number of violins.

The equation y⁻¹ = log₂(x)/10 elucidates that the inverse function of y is based on the logarithmic relationship between the number of violins (x) and the perceived loudness factor. This inverse function reveals how many times louder a single violin would sound compared to the total number of violins.

Applying the Chain Rule to d(y⁻¹)/dx, we derive d(y⁻¹)/dx = -1/(10x * ln(2)), showcasing the rate of change of the reciprocal of violins concerning the change in the perceived loudness factor. This derivative illustrates the sensitivity of how adjustments in the perceived loudness factor influence changes in the number of violins.