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Chapter 2: Problem 6

The profit \(P\), in thousands of dollars, that a manufacturer makes is a function of the number \(N\) of items produced in a year, and the formula is $$ P=-0.2 N^{2}+3.6 N-9 . $$ a. Express using functional notation the profit at a production level of 5 items per year, and then calculate that value. b. Determine the two break-even points for this manufacturer-that is, the two production levels at which the profit is zero. c. Determine the maximum profit if the manufacturer can produce at most 20 items in a year.

Short Answer

Expert verified
a. \(P(5) = 4\) thousand dollars. b. Break-even points are \(N = 6\) and \(N = 7.5\). c. Maximum profit is \(7.2\) thousand dollars.

Step by step solution

01

Express Profit Function at N=5

To find the profit at a production level of 5 items, substitute \(N = 5\) into the profit function. This gives\[P(5) = -0.2(5)^2 + 3.6(5) - 9.\]
02

Calculate Profit for N=5

Calculate the expression from Step 1:1. Compute \((5)^2 = 25\).2. Multiply \(-0.2 \times 25 = -5\).3. Compute \(3.6 \times 5 = 18\).4. Add these results: \(-5 + 18 - 9 = 4\).Thus, \(P(5) = 4\).
03

Set Function to Zero for Break-even

To find the break-even points, set the profit function equal to zero:\[-0.2 N^2 + 3.6 N - 9 = 0.\]
04

Solve Quadratic Equation

Use the quadratic formula \(N = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\) with \(a = -0.2\), \(b = 3.6\), \(c = -9\):1. Compute the discriminant: \(b^2 - 4ac = (3.6)^2 - 4(-0.2)(-9)\).2. Simplify: \(12.96 - 7.2 = 5.76\).3. Calculate \(N = \frac{{-3.6 \pm \sqrt{5.76}}}{-0.4}\).4. Simplify to find roots \(N_1 = 7.5\) and \(N_2 = 6\).
05

Optimize for Maximum Profit

The maximum profit occurs at the vertex of the parabola, given by \(N = -\frac{b}{2a}\):1. Calculate \(N = -\frac{3.6}{2(-0.2)} = 9\).2. Substitute \(N = 9\) into the profit function to find maximum profit:\[P(9) = -0.2(9)^2 + 3.6(9) - 9\].
06

Calculate Maximum Profit

Compute profit at \(N = 9\):1. \(9^2 = 81\), so \(-0.2 \times 81 = -16.2\).2. \(3.6 \times 9 = 32.4\).3. \(-16.2 + 32.4 - 9 = 7.2\).Thus, the maximum profit is \(7.2\) thousand dollars.

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Key Concepts

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

Break-even Points
Break-even points are the production levels at which a company's profit equals zero. These are critical for businesses because they represent the minimum level of sales needed to cover all expenses, without making any loss or gain.
To find the break-even points, you set the profit function to zero. For this specific profit function:
  • The equation is set as \[-0.2N^2 + 3.6N - 9 = 0.\]
  • This represents a quadratic equation, which typically has two solutions and thus two break-even points.
  • Solving this helps identify the number of items that need to be produced and sold for the profit to be exactly zero.
Understanding these points is crucial as they serve as thresholds that can guide business decisions related to scaling production.
Quadratic Formula
The quadratic formula is used to solve quadratic equations of the form \(ax^2 + bx + c = 0\). It allows you to find the values of \(N\) for which the expression equals zero.
In the format \(N = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\), you:
  • First calculate the discriminant \(b^2 - 4ac\), which indicates the number of possible solutions.
  • A positive discriminant suggests two real solutions, and a negative one means there are no real solutions.
  • Using this formula provides exact solutions for \(N\), representing critical values like break-even points.
For the equation \(-0.2N^2 + 3.6N - 9 = 0\), the quadratic formula gives the two break-even production levels as \(N = 7.5\) and \(N = 6\). This means the production needs to be between these values initially for the profit to grow from zero.
Maximum Profit
Maximum profit is the highest possible profit a company can achieve given certain constraints, such as production capacity. In mathematical terms, for a quadratic profit function, the maximum occurs at the vertex of the parabola.
To determine the maximum profit:
  • Find the vertex of the parabola, which for a quadratic function \(ax^2 + bx + c\) is given by \(N = -\frac{b}{2a}\).
  • Substitute the value of \(N\) from the vertex into the profit function to calculate the maximum profit.
  • In specific cases like this exercise, constraints like a maximum production of 20 items limit where this maximum may occur.
For our function, the vertex is at \(N = 9\), and inserting this value back into the equation produces a maximum profit of \(7.2\) thousand dollars.
Vertex of a Parabola
The vertex of a parabola is a crucial point as it represents the minimum or maximum value of a quadratic function. In the context of a profit function, it indicates either the maximum profit or minimum loss.
For a downward-opening parabola represented by a quadratic function \(ax^2 + bx + c\), the vertex is calculated as:
  • \(N = -\frac{b}{2a}\).
  • Substituting \(N\) back into the function provides the maximum value of the function.
  • It's important to consider the direction of the parabola opening to understand whether this vertex represents a maximum or minimum value.
In our example, the parabola opens downwards as indicated by the negative \(a\)-value, hence, the vertex at \(N = 9\) signals the point of maximum profit. Recognizing and calculating the vertex is essential for effective management and planning in production-related businesses.

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