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Chapter 2: Problem 2

The weekly profit \(P\) for a widget producer is a function of the number \(n\) of widgets sold. The formula is $$ P=-2+2.9 n-0.3 n^{2} $$ 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 7 thousand widgets sold. a. Make a graph of \(P\) versus \(n\). b. Calculate \(P(0)\) and explain in practical terms what your answer means. c. At what sales level is the profit as large as possible?

Short Answer

Expert verified
a. Plot \( P(n) = -2 + 2.9n - 0.3n^2 \) from 0 to 7. b. \( P(0) = -2 \), which means a $2,000 loss with no sales. c. Profit is maximized at about 4,833 widgets sold.

Step by step solution

01

Identify the function

The given profit function is \[ P = -2 + 2.9n - 0.3n^2 \] where \( P \) represents profit in thousands of dollars, and \( n \) is the number of widgets sold in thousands.
02

Graphing the function

To graph the function \( P(n) = -2 + 2.9n - 0.3n^2 \), you should select a range of \( n \) values from 0 to 7 (as the formula is valid up to 7 thousands of widgets) and compute the corresponding \( P \) values. Plot these points \((n, P)\) on a graph to visualize the quadratic curve.
03

Calculate P(0)

Substitute \( n = 0 \) into the profit function: \[ P(0) = -2 + 2.9(0) - 0.3(0)^2 = -2 \]This result implies that if no widgets are sold, the company incurs a loss of $2,000, represented by \( P(0) = -2 \).
04

Determine maximum profit

The maximum profit (vertex of a downward-opening parabola) can be found using the vertex formula for a quadratic function \( n = -\frac{b}{2a} \). Here, \( a = -0.3 \) and \( b = 2.9 \), so: \[ n = -\frac{2.9}{2(-0.3)} = \frac{2.9}{0.6} = 4.8333 \]Thus, the maximum profit occurs when \( n = 4.83 \), or when approximately 4,833 widgets are sold.

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Key Concepts

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

Profit Maximization
Profit maximization is a key concept in economics that businesses aim to achieve. When dealing with quadratic functions in real-world scenarios, such as the profit function of a company, understanding how to maximize profit becomes crucial. In the given exercise, we have a quadratic profit function: \[ P = -2 + 2.9n - 0.3n^2 \]This function represents the company's profit in thousands of dollars with respect to the number of widgets sold, measured in thousands.
To find the level of widget sales where profit is maximized, we need to locate the vertex of the parabola, as it represents either a maximum or minimum point. In this case, since the parabola opens downwards (indicated by the negative coefficient of the \(n^2\) term), the vertex will be the point of maximum profit.
The calculation showed that the maximum profit of roughly $4,833, or 4.83 thousand widgets sold, highlights the importance of analyzing the quadratic model to achieve the best business outcomes.
Graphing Quadratic Functions
Graphing quadratic functions helps visualize profit scenarios over different levels of widget sales. With the profit function \( P = -2 + 2.9n - 0.3n^2 \), we graph it to see how profit changes as sales change.
Begin by choosing values of \( n \) (from 0 up to 7, given the exercise constraints) and calculate the corresponding \( P \) values. Here, the vertex and the shape of the graph are especially important:
  • The vertex shows the maximum profit point.
  • The graph is a parabola that opens downwards, indicated by the negative coefficient of the squared term.
Plot these points on the graph to form the curve.
This visualization allows us to see not only where the maximum profit occurs, but also how profit behaves as sales increase or decrease. It's clear that there is an optimal point near \( n = 4.83 \) where profit peaks, aligning with the vertex found in calculations.
Vertex of a Parabola
The vertex of a parabola plays a crucial role in analyzing quadratic functions, particularly in identifying maximum or minimum values. This is especially useful in the context of profit maximization, as it allows businesses to determine the most profitable sales level.
For a quadratic function in the form \( ax^2 + bx + c \), the x-coordinate of the vertex can be found using the formula:\[ n = -\frac{b}{2a} \]In the given problem, substituting \( a = -0.3 \) and \( b = 2.9 \) into this formula gives us:
\[ n = -\frac{2.9}{2 \times (-0.3)} = \frac{2.9}{0.6} = 4.8333 \]This calculation shows that the vertex occurs when approximately 4,833 widgets are sold, representing the sales level at which the profit is maximized.
Understanding the vertex of a quadratic equation allows businesses to precisely tune their strategies and operations to maximize profitability based on sales data.

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