Problem 103 - Algebra and Trigonometry Real Mathematics, Real People

Question

Environment The predicted cost \(C\) (in thousands of dollars) for a company to remove \(p \%\) of a chemical from its wastewater is given by the model $$C=\frac{120 p}{10,000-p^{2}}, \quad 0 \leq p < 100$$ Write the partial fraction decomposition of the rational function. Verify your result by using the table feature of a graphing utility to create a table comparing the original function with the partial fractions.

Step-by-Step Solution

Answer

The partial fraction decomposition of \(C=\frac{120 p}{10,000-p^{2}}\) is \(C = \frac{60}{100 - p} + \frac{60}{100 + p}\).

Step 1: Identify the denominator and factor

The denominator of the function to decompose is \(10000 - p^{2}\). This can be factored using the difference of squares formula. The factored denominator is: \(100^2 - p^2 = (100 - p)(100 + p)\).

Step 2: Setup the partial fraction decomposition

Once the denominator is factored, you can setup the partial fraction decomposition. We will express \(C=\frac{120 p}{10,000-p^{2}}\) as \(C = \frac{A}{100 - p} + \frac{B}{100 + p}\).

Step 3: Find the values of A and B

Multiply through by the common denominator to get rid of the fractions: \(120p = A(100 + p) + B(100 - p)\). By comparing coefficients, A has to be 60 and B also has to be 60. Our decomposition now becomes \(C = \frac{60}{100 - p} + \frac{60}{100 + p}\).

Step 4: Verification

This can be verified by comparing a table of values for the original function and the partial fractions using a graphing utility, for different values of \(p\). If the values match for all values of \(p\), the decomposition is correct.

Key Concepts

Difference of Squares

Understanding the difference of squares is crucial when dealing with this type of problem. It refers to a specific algebraic expression: \(a^2 - b^2\). These expressions can be rewritten as \((a - b)(a + b)\), making them easier to work with in various mathematical problems. In the original exercise, the denominator \(10000 - p^2\) is a perfect candidate for this technique. Here, it can be factored into \((100 - p)(100 + p)\), simplifying the rational function. Recognizing and applying the difference of squares can transform complex expressions and lay the groundwork for partial fraction decomposition. By breaking them into simpler binomials, you make solving equations more approachable. This will aid in solving many algebraic problems efficiently.

Graphing Utility

A graphing utility can be a powerful tool when it comes to verifying computations like partial fraction decomposition. With such utilities, you can easily visualize functions and their components. In this exercise, using the graphing utility means plotting both the original rational function \(C = \frac{120p}{10000 - p^2}\) and its decomposed form.
Creating a table of values for both functions can prove that they are equivalent by showing that they produce the same output for corresponding inputs. This matching process confirms the accuracy of the decomposition and provides visual proof that the mathematical manipulations have been executed correctly. Software like Desmos or graphing calculators are ideal for these tasks, offering easy-to-use interfaces to support your learning.

Rational Function

Rational functions are expressions that involve the division of two polynomials. They represent ratios of polynomials and can exhibit a variety of behaviors based on their specific forms. In the given problem, the function \(C = \frac{120p}{10000 - p^2}\) is a rational function with a quadratic polynomial in its denominator and a linear term in its numerator.
Behavior such as asymptotes and intercepts depends on the degrees of these polynomials. For instance, vertical asymptotes arise where the denominator equals zero. Here, these occur at \(p = 100\) and \(p = -100\), the points where the denominator factors equal zero.
Recognizing the structure of rational functions assists in tasks like finding domain restrictions or potential simplifications, enriching your understanding of complex equations.

Coefficient Comparison

Coefficient comparison is a technique used to solve for unknown values in equations, especially in partial fraction decomposition. It involves aligning like terms from both sides of an equation to find coefficients of the polynomial terms. In this exercise, after setting up the partial fraction decomposition, the step is crucial for determining the constants \(A\) and \(B\) in the equation:
\(120p = A(100 + p) + B(100 - p)\).
By setting up equations for the coefficients of \(p\) and the constant terms separately, we can solve for \(A\) and \(B\). This requires isolating terms and ensuring both sides of the equation contribute equivalently to each component. Through this straightforward comparison, you'll frequently unravel segments of more intricate algebraic processes, enhancing your problem-solving toolkit. In this task, \(A = 60\) and \(B = 60\), making the decomposition accurate and verifiable.