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Chapter 1: Problem 25

Use your grapher to find the breakeven quantities for the given profit functions and the value of \(x\) that maximizes the profit. $$ P(x)=-4 x^{2}+38.4 x-59.36 $$

Short Answer

Expert verified
Breakeven at \( x \approx 1.94 \) and \( x \approx 7.66 \). Max profit at \( x = 4.8 \).

Step by step solution

01

Understand the Problem

We need to find the breakeven quantities and the value of \( x \) that maximizes the profit from the given profit function \( P(x) = -4x^2 + 38.4x - 59.36 \). The breakeven quantities are the values of \( x \) where profit \( P(x) = 0 \). The maximum profit occurs at the vertex of the parabola since this is a quadratic function.
02

Breakeven Analysis

To find the breakeven points, we need to set the profit equation equal to zero and solve for \( x \):\[ -4x^2 + 38.4x - 59.36 = 0 \]Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = -4 \), \( b = 38.4 \), and \( c = -59.36 \).
03

Solve Using the Quadratic Formula

First, calculate the discriminant:\[ b^2 - 4ac = (38.4)^2 - 4(-4)(-59.36) \]\[ = 1474.56 - 949.76 = 524.8 \]Now, use the quadratic formula:\[ x = \frac{-38.4 \pm \sqrt{524.8}}{-8} \]
04

Calculate Roots

First, find the square root of the discriminant:\[ \sqrt{524.8} \approx 22.9 \]Substitute back into the quadratic formula for each root:\[ x_1 = \frac{-38.4 + 22.9}{-8} = 1.9375 \]\[ x_2 = \frac{-38.4 - 22.9}{-8} = 7.6625 \]These are the breakeven quantities.
05

Find Maximum Profit

The maximum profit occurs at the vertex of the parabola. The \( x \)-coordinate of the vertex is found by:\[ x_{vertex} = \frac{-b}{2a} = \frac{-38.4}{2(-4)} = \frac{38.4}{8} = 4.8 \]
06

Conclusion

The breakeven quantities are approximately \( x = 1.94 \) and \( x = 7.66 \). The value of \( x \) that maximizes the profit is \( 4.8 \).

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Key Concepts

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

Breakeven Analysis
Breakeven analysis helps businesses determine when they will start to make a profit. In simple terms, it is the point where costs equal revenues, resulting in zero profit or loss. For our quadratic function \( P(x) = -4x^2 + 38.4x - 59.36 \), the goal is to find the values of \( x \) where \( P(x) = 0 \). These are known as the breakeven points.
  • Set the profit function equal to zero: \(-4x^2 + 38.4x - 59.36 = 0\).
  • Solve this equation to find the breakeven quantities.
Breakeven points tell us at which quantity of goods sold can cover total costs, with neither profit nor loss, a crucial insight for economic sustainability.
Profit Maximization
Profit maximization entails identifying the level of sales that yields the highest possible profit. For quadratic profit functions like \(P(x) = -4x^2 + 38.4x - 59.36\), profit is maximized at the vertex of the parabola. Why? The nature of a downward-opening parabola means it reaches its peak at the top, which represents maximum profit.
  • Calculate the vertex using \(x = \frac{-b}{2a}\).
  • This \(x\) value provides the quantity at which maximum profit is achieved.
In this specific problem, the maximum profit occurs when \(x = 4.8\), indicating the ideal quantity for profit maximization.
Quadratic Formula
The quadratic formula is a powerful tool used to solve quadratic equations of the form \(ax^2 + bx + c = 0\). It is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula helps find the roots of the quadratic equation, which are crucial in breakeven analysis and profit maximization.
  • Calculate the discriminant, \(b^2 - 4ac\), to determine the nature of the roots (real or complex).
  • Use the formula to solve for \(x\), providing essential values like breakeven points in this context.
In our exercise, this formula helps explore where our profit function intersects the axis for breakeven.
Vertex of a Parabola
A parabola's vertex represents either the highest or lowest point of the curve. For a downward-facing parabola, it is the top point, symbolizing maximum value, such as profit.
  • The vertex can be calculated using \(x = \frac{-b}{2a}\) from the quadratic equation \(ax^2 + bx + c\).
  • It gives insight into optimal points for functions like cost, revenue, or profit.
In the function \(P(x) = -4x^2 + 38.4x - 59.36\), the vertex is at \(x = 4.8\), pinpointing where profit peaks. Knowing the vertex is essential for making strategic business decisions on output and pricing.

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Most popular questions from this chapter

One bank advertises a nominal rate of \(6.5 \%\) compounded quarterly. A second bank advertises a nominal rate of \(6.6 \%\) compounded daily. What are the effective yields? In which bank would you deposit your money?

Elliott \(^{53}\) studied the temperature affects on the alder fly. He showed that in a certain study conducted in the laboratory, the number of pupae that successfully completed pupation \(y\) was approximately related to temperature \(t\) in degrees Celsius by \(y=-0.718 t^{2}+21.34 t-112.42\) a. Determine the temperature at which this model predicts the maximum number of successful pupations. b. Determine the two temperatures at which this model predicts that there will be no successful pupation.

The human population of the world was about 6 billion in the year 2000 and increasing at the rate of \(1.3 \%\) a year \(^{67}\) Assume that this population will continue to grow exponentially at this rate, and determine the population of the world in the year 2010 .

Suppose the demand equation for a commodity is of the form \(p=m x+b\), where \(m<0\) and \(b>0\). Suppose the cost function is of the form \(C=d x+e\), where \(d>0\) and \(e<0 .\) Show that profit peaks before revenue peaks.

Using your grapher, graph on the same screen \(y_{1}=\) \(f(x)=|x|\) and \(y_{2}=g(x)=\sqrt{x^{2}} .\) How many graphs do you see? What does this say about the two functions?
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