Question

sound intensity, \\(i\\), from a spherical source is a function of the distance, \\(r\\), from the source of the sound. it is represented by the function

\\i = \frac{p}{4\pi r^2}\\

where \\(p\\) is the power of the sound. explain the behavior of the graph of \\(i\\) and what it means in context.

Explanation:

Analyze the function type

Using the Rational Functions knowledge point

$$ I(r) = \frac{P}{4\pi r^2},\quad r > 0,\quad P > 0 $$

Determine graphical behavior

Using the Rational Function Graphing knowledge point

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Interpret in physical context

As distance \(r\) from the source increases, sound intensity \(I\) decreases rapidly following an inverse-square relationship. Very close to the source (\(r \to 0\)), the intensity becomes extremely high, while at very large distances (\(r \to \infty\)), the sound fades away toward zero intensity.

Answer:

The graph of \(I = \frac{P}{4\pi r^2}\) (for \(r > 0\)) is strictly decreasing, with a vertical asymptote at \(r = 0\) and a horizontal asymptote at \(I = 0\).

In context, this means that as you move further away from the sound source, the sound intensity decreases rapidly according to the inverse-square law. Extremely close to the source, the sound is intensely loud, while at very great distances, the sound intensity approaches zero and becomes nearly silent.