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Chapter 2: Problem 2

The monthly profit \(P\) for a widget producer is a function of the number \(n\) of widgets sold. The formula is $$ P=-15+10 n-0.2 n^{2} . $$ Here \(P\) is measured in thousands of dollars, \(n\) is measured in thousands of widgets, and the formula is valid up to a level of 15 thousand widgets sold. a. Make a graph of \(P\) versus \(n\). b. Calculate \(P(1)\) and explain in practical terms what your answer means. c. Is the graph concave up or concave down? Explain in practical terms what this means. d. The break-even point is the sales level at which the profit is 0 . Find the break-even point for this widget producer.

Short Answer

Expert verified
a) Plot \( P = -15 + 10n - 0.2n^2 \); b) \( P(1) = -5.2 \), a loss of $5,200; c) Concave down, diminishing returns; d) Break-even at 1,605 and 9,395 widgets sold.

Step by step solution

01

Graphing the Function P(n)

To graph the function, we need to understand the equation: \( P = -15 + 10n - 0.2n^2 \). This is a quadratic function in standard form \( ax^2 + bx + c \) where \( a = -0.2 \), \( b = 10 \), and \( c = -15 \). It is a parabola opening downward because \( a < 0 \). We can plot values of \( n \) from 0 to 15 (in thousands) to see the parabola shape and the profit over this range.
02

Calculate P(1)

To find \( P(1) \), substitute \( n = 1 \) into the equation: \( P(1) = -15 + 10(1) - 0.2(1)^2 \). Simplify to get \( P(1) = -15 + 10 - 0.2 = -5.2 \). Since \( P \) is in thousands, this means the profit is \(-5,200\) dollars. The company incurs a loss of \$5,200 when 1,000 widgets are sold.
03

Determine Concavity

To determine concavity, evaluate the second derivative of \( P(n) \). The first derivative is \( P'(n) = 10 - 0.4n \). The second derivative is \( P''(n) = -0.4 \). Since \( P''(n) < 0 \), the graph is concave down. This means that as more widgets are sold, the increase in profit gets smaller, indicating diminishing returns per additional widget.
04

Finding the Break-even Point

The break-even point is where \( P(n) = 0 \). Setting the equation to zero gives \( 0 = -15 + 10n - 0.2n^2 \). Solving \(-0.2n^2 + 10n - 15 = 0\) using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), we find \( n = \frac{-10 \pm \sqrt{100 + 12}}{-0.4} \). Solving for the roots gives \( n \approx 1.605 \) and \( n \approx 9.395 \). Thus, 1,605 and 9,395 are the approximate break-even points (in thousands), or 1,605 and 9,395 widgets.

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Key Concepts

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

Profit Maximization
Every business aims to achieve maximum profit, and understanding quadratic functions can help with this. In the quadratic function provided, \( P = -15 + 10n - 0.2n^2 \), \( P \) stands for profit in thousands of dollars, and \( n \) is the number of thousands of widgets sold. This function shows that profit increases with each widget sold but at a decreasing rate due to the negative \( n^2 \) term. It also implies a maximum profit will be reached before the profit starts declining. This point is called the vertex of the parabola.
To find the number of widgets sold for maximum profit, use the formula for the vertex of a parabola \( n = -\frac{b}{2a} \), where \( a = -0.2 \) and \( b = 10 \). Solving yields \( n = \frac{10}{0.4} = 25 \). However, since the formula applies up to 15,000 widgets, the company achieves maximum profit beyond this range, and practical evaluation must be within limitations.
Concavity
Concavity tells us how the graph of a function curves. In this context, because the quadratic coefficient \( a \) is negative in the profit function, the graph is concave down. When the graph is concave down, it signifies the profit increases as more widgets are sold but at a decreasing rate.
The second derivative \( P''(n) = -0.4 \) confirms concave down, as \( P''(n) < 0 \). This reflects diminishing returns 鈥� every additional widget sold contributes less to the profit than the previous one. Understanding concavity helps businesses recognize when further production may yield less financial benefit and informs decisions on the optimal quantity of widgets to produce.
Break-even Point
The break-even point is the sales level where the revenue equals costs, resulting in zero profit. In other words, it's the point where the profit function crosses the horizontal axis. For the widget producer, solving the function \( 0 = -15 + 10n - 0.2n^2 \) using the quadratic formula helps find these critical points.
  • The quadratic formula: \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -0.2 \), \( b = 10 \), and \( c = -15 \).
  • Solving gives two values: approximately \( 1.605 \) and \( 9.395 \).
These translate to around 1,605 and 9,395 widgets required to break even. Selling widgets below these points results in a loss, while selling beyond the highest break-even point results in a profit. These indicators are crucial for financial decision-making and planning.
Graphing Functions
Graphing functions provides a visual representation, making it easier to understand relationships between variables. For the quadratic profit function \( P = -15 + 10n - 0.2n^2 \), the graph is a downward-opening parabola.
Graphs help:
  • Visualize maximum and minimum values quickly, aiding in identifying the vertex.
  • Observe the concavity to understand the profit behavior as sales volume changes.
  • Identify intercepts, such as the break-even points, more intuitively.
Plotting values of \( n \) from 0 to 15 on the x-axis (thousand widgets) and corresponding \( P \) on the y-axis (thousand dollars) will help chart the path to optimal production levels and economic insights for strategic decisions.
Each point on the graph reflects a different sales level and its financial outcome, unlocking a deeper understanding of sales strategies.

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