Problem 41 - Algebra A Combined Function

Question

If the cost, \(C,\) for manufacturing \(x\) units of a certain product is given by \(C=x^{2}-15 x+50,\) find the number of units manufactured at a cost of \(\$ 9500\).

Step-by-Step Solution

Answer

105 units are manufactured.

Step 1: Set Up the Equation

Begin by setting up the equation given in the problem. We know that the cost function is given by \(C = x^2 - 15x + 50\). We need to find the value of \(x\) when \(C = 9500\), so substitute \(9500\) into the equation: \[ 9500 = x^2 - 15x + 50 \]

Step 2: Rearrange the Equation

Rearrange the equation into standard quadratic form by moving all terms to one side: \[ x^2 - 15x + 50 - 9500 = 0 \] Which simplifies to: \[ x^2 - 15x - 9450 = 0 \]

Step 3: Solve the Quadratic Equation

We have a quadratic equation \(x^2 - 15x - 9450 = 0\). We will use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = -15\), and \(c = -9450\). Substitute these values into the formula: \[ x = \frac{-(-15) \pm \sqrt{(-15)^2 - 4\cdot1\cdot(-9450)}}{2\cdot1} \] Which simplifies to: \[ x = \frac{15 \pm \sqrt{225 + 37800}}{2} \]

Step 4: Calculate the Discriminant

First, calculate the value under the square root (the discriminant): \(\sqrt{38025}\). Calculating gives us \(\sqrt{38025} = 195\).

Step 5: Solve for x

Substitute \(\sqrt{38025} = 195\) back into the formula to solve for \(x\): \[ x = \frac{15 \pm 195}{2} \] This yields two possible solutions for \(x\): \[ x = \frac{15 + 195}{2} \Rightarrow x = 105 \] \[ x = \frac{15 - 195}{2} \Rightarrow x = -90 \]

Step 6: Interpret the Results

Since the number of units (\(x\)) cannot be negative, we discard \(x = -90\). Thus, the number of units manufactured is \(x = 105\).

Key Concepts

Understanding Cost Function in Manufacturing

To begin with, a cost function is a mathematical formula that helps you calculate the total cost incurred in manufacturing a certain number of units. A simple representation is based on a quadratic equation, often taking the form \( C = ax^2 + bx + c \). Here, \( x \) represents the number of units produced, while \( a \), \( b \), and \( c \) are constants that represent different aspects like variable costs, fixed costs, and other incremental costs.

• A cost function helps businesses plan and forecast expenses.
• Understanding the function allows you to find the cost for producing any number of units.
• It is crucial for determining profitability and setting pricing strategies.

In the original exercise, the cost function \( C = x^2 - 15x + 50 \) is used. This equation calculates the cost for manufacturing \( x \) units. The task is to find out how many units are produced when the cost equals $9500.
This involves solving the quadratic equation for an unknown \( x \), where \( C \) is already given.

Demystifying the Quadratic Formula

Quadratic equations are a type of polynomial and are often shown in the standard form \( ax^2 + bx + c = 0 \). When we're dealing with cost functions or other real-world scenarios, solving these equations is key. The quadratic formula is a reliable way to find the values of \( x \) that satisfy the quadratic equation. It looks like this:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

• \(-b\): Involves changing the sign of the \(b\) term.
• \(\pm\): Indicates there are two possible solutions for \(x\).
• \(\sqrt{b^2 - 4ac}\): Known as the discriminant, it determines the nature of the roots.
• \(2a\): The denominator affecting the division and scaling of results.

In our exercise, the formula helps to solve \( x^2 - 15x - 9450 = 0 \) for \( x \), providing insight into how many units were required to reach a specific cost.

Decoding the Discriminant

The discriminant is the part of the quadratic formula under the square root: \( b^2 - 4ac \). It plays a vital role as it tells us about the nature of the roots.

• If the discriminant is positive, we have two distinct real solutions.
• If it is zero, there is exactly one real solution.
• If the discriminant is negative, the solutions are complex and not real numbers.

For the equation in the current problem, the discriminant calculation is \( 225 + 37800 \), leading to \( 38025 \). A positive value signals two potential real values for \( x \). This helps in identifying viable solutions for the manufacturing count, eventually leading to the decision to discard negative outcomes due to physical impossibilities.

Applying Problem Solving Techniques

Problem-solving involves a series of logical steps to derive meaningful results from given data. When we approach exercises involving quadratic equations, several problem-solving stages are necessary:

• Set up the equation based on the problem's context.
• Rearrange it into standard quadratic form for ease of solving.
• Use the quadratic formula to find possible solutions.
• Analyze each potential solution to select the most feasible one.

In our specific case, interpreting the results correctly—discarding a negative solution—is pivotal for practical application. This approach not only leverages mathematical skills but also integrates real-world reasoning to arrive at an executable answer, like identifying the number of units manufactured when the given cost is $9500.