We are given the intensity function for sound intensity $I(r)$ as a function of the distance $r$ from a sound source, which is given by:

$I(r) = \frac{P}{4 \pi r^2}$

Where:

• $I(r)$ is the sound intensity (in watts per square meter, W/m²),
• $P$ is the acoustic power (in watts, W),
• $r$ is the distance from the sound source (in meters),
• $\pi$ is approximately 3.1416.

We will solve each part step by step.

Part a: Sound Intensity at $r_1 = 10 \, \text{m}$ and $r_2 = 15 \, \text{m}$

Given $P = 100 \, \text{W}$, the intensity function becomes:

$I(r) = \frac{100}{4 \pi r^2}$

1. At $r_1 = 10 \, \text{m}$:

$I(10) = \frac{100}{4 \pi (10^2)} = \frac{100}{4 \pi \cdot 100} = \frac{100}{400 \pi} = \frac{1}{4 \pi} \, \text{W/m}^2$ Approximating $\pi \approx 3.1416$:

$I(10) \approx \frac{1}{4 \times 3.1416} \approx \frac{1}{12.5664} \approx 0.0796 \, \text{W/m}^2$

2. At $r_2 = 15 \, \text{m}$:

$I(15) = \frac{100}{4 \pi (15^2)} = \frac{100}{4 \pi \cdot 225} = \frac{100}{900 \pi} = \frac{1}{9 \pi} \, \text{W/m}^2$ Approximating $\pi$:

$I(15) \approx \frac{1}{9 \times 3.1416} \approx \frac{1}{28.2744} \approx 0.0354 \, \text{W/m}^2$

Slope of the Secant Line

The slope $m$ of the secant line through the points $(r_1, I(r_1))$ and $(r_2, I(r_2))$ is given by the formula:

$m = \frac{I(r_2) - I(r_1)}{r_2 - r_1}$

Substitute the values for $I(10)$, $I(15)$, $r_1 = 10$, and $r_2 = 15$:

$m = \frac{0.0354 - 0.0796}{15 - 10} = \frac{-0.0442}{5} = -0.00884 \, \text{W/m}^2 \text{ per meter}$

Interpretation of the Slope:

The negative slope indicates that the intensity decreases as the distance from the source increases, which is consistent with the inverse-square law of sound intensity. The rate of decrease in intensity is approximately $0.00884 \, \text{W/m}^2$ per meter for the given range.

Part b: General Slope of the Secant Line for Any $r_1$ and $r_2$

The formula for the sound intensity is:

$I(r) = \frac{100}{4 \pi r^2}$

For two arbitrary points $r_1$ and $r_2$, the slope of the secant line is:

$m = \frac{I(r_2) - I(r_1)}{r_2 - r_1}$

Substituting the intensity function:

$m = \frac{\frac{100}{4 \pi r_2^2} - \frac{100}{4 \pi r_1^2}}{r_2 - r_1}$

Factor out $\frac{100}{4 \pi}$:

$m = \frac{100}{4 \pi} \cdot \frac{\frac{1}{r_2^2} - \frac{1}{r_1^2}}{r_2 - r_1}$

This is the general formula for the slope of the secant line between any two points on the graph of the sound intensity function.

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Would you like further details on any step? Here are 5 related questions for more exploration:

1. What is the physical meaning of the inverse-square law in sound propagation?
2. How does the secant line slope change for different distances $r_1$ and $r_2$?
3. How is sound intensity affected by changes in acoustic power $P$?
4. How would the sound intensity change if the medium affected the propagation?
5. Can the slope of the secant line be used to estimate the sound intensity decay rate?

Tip: The inverse-square law is a principle that applies to many physical phenomena, including light and gravitational forces, not just sound.