Wind Turbine Noise Level Analysis
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Questions and Answers

What is the value of the constant a in the noise level function S(d) when d = 0?

  • 0
  • 105 (correct)
  • 100
  • 49

What is the approximate value of b in the function S(d) modeled by S(d) = abd?

  • 0.992408 (correct)
  • 2.0
  • 1.1
  • 0.5

What is the average rate of change of the noise level from d = 0 to d = 100 meters?

  • 1.1 decibels per meter
  • -1.0 decibels per meter
  • -0.56 decibels per meter (correct)
  • 0.5 decibels per meter

If the average rate of change is -0.56 decibels per meter, what would the estimated noise level be at d = 120 meters?

<p>37.8 decibels (D)</p> Signup and view all the answers

What equation must be solved to find the distance m where the noise level is 20 decibels?

<p>20 = 105(0.992408)<sup>m</sup> (B)</p> Signup and view all the answers

What is the value of (g o h)(2)?

<p>3 (B)</p> Signup and view all the answers

What does a negative average rate of change indicate in the context of noise level from the turbine?

<p>Noise level is decreasing with distance (C)</p> Signup and view all the answers

If the noise level at d = 100 meters is 49 decibels, what is the sound level at this distance compared to d = 0 meters?

<p>It has decreased significantly (C)</p> Signup and view all the answers

Which of the following represents the expression for j(h(x)) simplified?

<p>-4x^2 + 20x - 24 (D)</p> Signup and view all the answers

Considering the noise level function S(d) = 105(0.992408)d, what happens to the noise level as d increases?

<p>Noise level decreases gradually (C)</p> Signup and view all the answers

What is the inverse function f-1(x) when f(x) = (x - 4)2 + 3?

<p>√(x - 3) + 4 (A)</p> Signup and view all the answers

For what values of x does f-1(f(x)) = x hold true?

<p>x ≥ 3 (D)</p> Signup and view all the answers

What is the equation of the exponential function that goes through the points (1, 8) and (2, 4)?

<p>f(x) = 16(1/2)<sup>x</sup> (D)</p> Signup and view all the answers

What is the value of x that satisfies the equation 9-x+15 = 27x?

<p>6 (A)</p> Signup and view all the answers

What value of 'a' results in the graph of f o g crossing the y-axis at 23?

<p>3 (B), -3 (C)</p> Signup and view all the answers

What is the correct simplified form of (j o h)(x)?

<p>-4x<sup>2</sup> + 20x - 24 (D)</p> Signup and view all the answers

Flashcards

Average Rate of Change

The rate of change of a function over a specific interval. It measures how much the output changes for each unit change in the input.

Inverse Square Law

A function that models the relationship between the distance from a source and the intensity of a phenomenon. It often involves an exponential decay.

Equation from Data

An equation derived from the given information that relates the variables in the problem.

Linear Approximation

A technique that uses a known point and the average rate of change to estimate the function's value at a nearby point.

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Exponential Function

A type of function that can be written in the form abd, where a and b are constants and d is the independent variable.

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Finding Constants

The process of finding the values of the constants in a function using the given data points.

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Solving for x

A way to find the value of the independent variable that leads to a specific output value in the function.

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Interpreting Results

The process of understanding and interpreting the results in the context of the problem.

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Function Composition

A mathematical operation that combines two functions by applying one function to the output of another. It is represented as (f o g)(x) = f(g(x)).

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Inverse Function

The inverse function of f(x) is a function that 'undoes' the operation of f(x). It is denoted as f-1(x). If you apply f(x) to a value and then apply its inverse, you get back the original value.

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Exponential Equation

A type of equation where the variable appears in the exponent. It often involves solving for the unknown base or exponent.

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Y-intercept

The point where a graph intersects the y-axis. This happens when the x-value is zero. It represents the value of the function when the input is zero.

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Composition of Functions (f o g)(x)

In this context, it refers to the result of applying one function to another. It is represented by (f o g)(x) = f(g(x)).

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Function f(x)

The function that takes a value as input and transforms it using a set of rules. It can be represented by a formula, graph or table.

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Domain of the Inverse Function

Values of x that satisfy the equation f-1(f(x)) = x or f(f-1(x)) = x. These are all the values that can be both the input of f(x) and output of f-1(x).

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Study Notes

Wind Turbine Noise Level

  • A wind turbine's blades generate noise that can be measured at different distances.
  • Noise level (S(d)) is modeled by S(d) = abd, where S(d) is the noise level in decibels at a distance of d meters from the turbine.
  • At 0 meters, the noise level is 105 decibels.
  • At 100 meters, the noise level is 49 decibels.

Equations for Noise Level

  • Using the given data, two equations can be constructed to find constants 'a' and 'b'.
  • 105 = ab0 (or simply 105 = a)
  • 49 = ab100

Determining Constants 'a' and 'b'

  • a = 105 (from the equation 105 = a)
  • b = 0.992408 (solved by calculating b = (49/105)^(1/100))

Average Rate of Change

  • The average rate of change of noise level between 0 and 100 meters is calculated using (S(100) - S(0)) / (100 - 0).
  • Average rate of change = -0.56 decibels per meter

Interpretation of Average Rate of Change

  • On average, the noise level decreases by 0.56 decibels for each additional meter as the distance from the turbine increases from 0 to 100 meters.

Estimating Noise Level at 120 Meters

  • Using the average rate of change, the noise level at 120 meters can be estimated.
  • Estimated noise level at 120 meters is approximately 37.8 decibels.

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Description

This quiz explores the noise levels generated by wind turbines at varying distances. It focuses on the equations used to model the noise level, the determination of constants 'a' and 'b', and the calculation of the average rate of change in noise level. Test your understanding of these concepts and their applications in real-world scenarios.

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