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Chapter 3: Problem 35

ENVIRONMENTAL SCIENCE: Pollution Two chemical factories are discharging toxic waste into a large lake, and the pollution level at a point \(x\) miles from factory A toward factory B is \(P(x)=3 x^{2}-72 x+576\) parts per million (for \(0 \leq x \leq 50)\). Find where the pollution is the least.

Short Answer

Expert verified
The pollution is least at 12 miles from factory A with 72 ppm.

Step by step solution

01

Identify the problem

We are given a pollution level function \( P(x) = 3x^2 - 72x + 576 \) and need to find the point \( x \) where this function reaches its minimum value, which indicates the least pollution.
02

Recognize the function type

The function \( P(x) = 3x^2 - 72x + 576 \) is a quadratic function of the form \( ax^2 + bx + c \), where \( a = 3 \), \( b = -72 \), and \( c = 576 \). Quadratic functions are parabolas and since \( a > 0 \), this parabola opens upwards, meaning it has a minimum point.
03

Finding the vertex of the parabola

The vertex of a parabola given by \( ax^2 + bx + c \) is found at \( x = -\frac{b}{2a} \). Substitute \( b = -72 \) and \( a = 3 \) into the formula: \( x = \frac{72}{2 \times 3} = 12 \). Thus, the minimum pollution level occurs at \( x = 12 \) miles from factory A.
04

Verify the result

Ensure that the value \( x = 12 \) is within the given range \( 0 \leq x \leq 50 \). Since 12 is within this range, the solution is valid.
05

Calculate pollution level at minimum

Substitute \( x = 12 \) into the function to find the pollution level: \( P(12) = 3(12)^2 - 72(12) + 576 \). Simplifying, \( P(12) = 3(144) - 864 + 576 \), which equals \( 72 \) parts per million. This confirms that the pollution is least at \( x = 12 \) miles.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Functions
Quadratic functions are a special kind of polynomial that take the form of \( ax^2 + bx + c \). These functions create curves known as parabolas when graphed on a coordinate plane. Quadratic functions are widespread in various applications, including physics, engineering, and economics, because they can model relationships that involve squared terms.
  • For a quadratic equation \( ax^2 + bx + c \), \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \).
  • If \( a > 0 \), the parabola opens upwards and has a minimum point, while if \( a < 0 \), it opens downwards and has a maximum point.
Understanding quadratic functions is crucial because they can depict various real-world scenarios, such as projectile motion and optimizing costs.
Vertex of Parabola
The vertex of a parabola plays a critical role in understanding the graph's behavior. Consider it the peak or the lowest point of the parabola, depending on whether it opens upwards or downwards.
  • The vertex's x-coordinate can be calculated using the formula \( x = -\frac{b}{2a} \) for a quadratic function \( ax^2 + bx + c \).
  • This point is vital for identifying the minimum or maximum value that the quadratic function attains.
In our pollution problem, finding the vertex helped us determine the point where pollution reached its minimum level. This approach is a powerful technique in optimization problems, allowing us to find the best possible outcome within a given context.
Environmental Pollution
Environmental pollution is a pressing issue, often resulting from human activity and industrial processes. In many scenarios, mathematical models like quadratic functions help in assessing and optimizing the impact.
  • Pollution levels in an ecosystem can be analyzed by mathematical expressions to determine areas of concern.
  • Mathematics aids in strategizing efficient resource management and reducing pollution exposure.
Understanding how to apply calculus and algebra to environmental issues can lead to more sustainable practices and policies, ensuring a healthier planet. In our example, using mathematics to find the least polluted point informs environmental strategies.
Mathematical Modelling
Mathematical modeling transforms real-world problems into mathematical language, providing solutions through analysis and computation. This process is beneficial for predicting outcomes and making informed decisions.
  • Models often employ equations like quadratic functions to portray physical, biological, or social systems.
  • With mathematical models, we can simulate scenarios and examine the effects of different variables.
By applying mathematical modeling to problems such as pollution, we can pinpoint crucial areas that require attention, optimize resources, and implement effective interventions. This kind of analysis not only provides insights but also aids in long-term planning and policy-making.

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