To address the questions related to the given problem:

Part 1: Understanding the Noise Level Function

The noise level $S(d)$ is described as $S(d) = a \cdot b^d$, where:

• $S(d)$ represents the noise level (in decibels) at distance $d$ meters.
• $a$ and $b$ are constants that define the relationship between the noise level and the distance.

We know:

• At $d = 0$, $S(0) = 105$.
• At $d = 100$, $S(100) = 49$.

These two data points allow us to calculate the constants $a$ and $b$.

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Step 1: Solving for $a$

When $d = 0$, $S(0) = a \cdot b^0 = a \cdot 1 = a$.
Thus, $a = 105$.

Step 2: Solving for $b$

Using the second data point, $S(100) = 49$: $49 = 105 \cdot b^{100}.$ Divide both sides by $105$: $b^{100} = \frac{49}{105} = \frac{7}{15}.$ To solve for $b$, take the 100th root: $b = \left(\frac{7}{15}\right)^{\frac{1}{100}}.$

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Part 2: Writing the General Form of $S(d)$

Substituting $a = 105$ and $b = \left(\frac{7}{15}\right)^{\frac{1}{100}}$, the noise level function is: $S(d) = 105 \cdot \left(\left(\frac{7}{15}\right)^{\frac{1}{100}}\right)^d.$ Simplify: $S(d) = 105 \cdot \left(\frac{7}{15}\right)^{\frac{d}{100}}.$

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Ready for Your Questions

Let me know if you'd like a step-by-step explanation or have specific questions! Here are five relative follow-up questions to deepen understanding:

1. How does the constant $a$ relate to the noise level at the source of the turbine?
2. Why does the value of $b$ decrease as distance $d$ increases?
3. How does the function $S(d)$ illustrate the inverse relationship between distance and noise?
4. Could the function $S(d)$ be used to estimate the noise level at 200 meters? How?
5. How would the noise level change if the turbine blades became quieter, altering the initial decibel reading?

Tip: When working with exponential decay models, identifying the constants accurately ensures the function closely represents real-world behavior.