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Chapter 5: Problem 30

The profit function for a company that manufactures cameras is \(\mathrm{P}(x)=-x^{2}+350 x-15,000 .\) Under present conditions, can the company achieve a profit of \(\$ 20,000 ?\) Use the discriminant to explain your answer.

Short Answer

Expert verified
The company cannot achieve a profit of \$20,000, as the discriminant is negative.

Step by step solution

01

Understand the Problem

We need to determine if the profit function \( P(x) = -x^2 + 350x - 15,000 \) can be equal to 20,000. This is done by checking for real solutions to \( P(x) = 20,000 \). We'll use the discriminant of the quadratic equation to decide this.
02

Set Up the Equation

To find when the profit equals \$20,000, we set up the equation \(-x^2 + 350x - 15,000 = 20,000\). Simplifying it, we obtain a new equation: \[-x^2 + 350x - 35,000 = 0\].
03

Determine the Discriminant

The quadratic equation is in the standard form \( ax^2 + bx + c = 0 \) where \( a = -1 \), \( b = 350 \), and \( c = -35,000 \). The discriminant \( D \) is given by the formula \( D = b^2 - 4ac \).
04

Calculate the Discriminant

Substituting the values of \( a \), \( b \), and \( c \) into the discriminant formula, we calculate:\[ D = 350^2 - 4(-1)(-35,000) \] \[ D = 122,500 - 140,000 \] \[ D = -17,500 \].
05

Analyze the Discriminant

Since the discriminant \( D = -17,500 \) is negative, this indicates there are no real solutions to the equation \(-x^2 + 350x - 35,000 = 0\). This means the profit cannot reach \$20,000 under current conditions.

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Key Concepts

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

Profit Function
A profit function is a mathematical representation that depicts how a firm's profit relates to the number of units it sells or produces. In this context, the company manufacturing cameras has a profit function:
  • It is given by the formula:\[ P(x) = -x^2 + 350x - 15,000 \]
  • Here, \( x \) represents the number of cameras produced and sold.
  • The equation showcases a maximum point where profit is maximized, as evident from the negative coefficient of \( x^2 \), forming a parabolic curve that opens downwards.
Understanding the profit function allows us to analyze the company's financial performance based on production levels. This model shows the point where further production might lead to a decrease in overall profit due to increasing costs.
Discriminant
The discriminant is a mathematical tool used to determine the nature of the roots of a quadratic equation. It is calculated from the coefficients of the quadratic equation in the form \( ax^2 + bx + c = 0 \) using the formula:
  • \[ D = b^2 - 4ac \]
The value of the discriminant gives insight into the solutions:
  • If \( D > 0 \), there are two distinct real solutions.
  • If \( D = 0 \), there is exactly one real solution.
  • If \( D < 0 \), there are no real solutions.
In our exercise, the discriminant was negative (\( D = -17,500 \)), implying that the equation has no real solutions. This indicates the profit cannot reach the desired amount of \(20,000\) under current circumstances.
Real Solutions
Real solutions refer to the actual values of \( x \) that satisfy a given equation within the realm of real numbers. In quadratic equations, checking for real solutions helps determine practical outcomes.
  • In our problem, discovering whether \( x \) values could make the profit function equal to 20,000 is vital.
  • Since the discriminant is negative, it signals that no real solution for \( x \) exists.
Without real solutions, we understand that the function cannot achieve the profit target due to the current ineffective business scenario or production constraints. Consequently, efforts could be directed towards changing factors such as cost structure or pricing strategy to achieve financial goals.
Quadratic Formula
The quadratic formula is a powerful method used to find the roots of a quadratic equation. Presented as:
  • \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This equation provides solutions based on the values of \( a \), \( b \), and \( c \) in the standard quadratic equation \( ax^2 + bx + c = 0 \).
  • The term under the square root, \( b^2 - 4ac \), is the discriminant and it reveals whether the solutions are real or complex.
  • In situations where the discriminant is negative (as in our camera manufacturing company's scenario), applying the quadratic formula confirms that the roots would be non-real.
Hence, understanding the quadratic formula supports identifying feasible solutions when dealing with profit or other algebraic functions in practical settings.

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Most popular questions from this chapter

In \(18-35,\) find each common solution algebraically. Express irrational roots in simplest radical form. $$ \begin{array}{l}{x^{2}+y^{2}=20} \\ {3 x-y=10}\end{array} $$

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The profit function, in thousands of dollars, for a company that makes graphing calculators is \(\mathrm{P}(x)=-5 x^{2}+5,400 x-106,000\) where \(x\) is the number of calculators sold in the millions. a. Graph the profit function \(\mathrm{P}(x)\) b. How many calculators must the company sell in order to make a profit?

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Write a quadratic equation with integer coefficients for each pair of roots. \(-3,4\)
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