Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 1: Problem 21
Find the value of \(x\) that maximizes the profit. Find the break-even quantities (if they exist); that is, find the value of \(x\) for which the profit is zero. Graph the solution. $$ R(x)=-x^{2}+30 x, C(x)=15 x+36 $$
Short Answer
Expert verified
The maximum profit is at \(x = 7.5\); break-even quantities are \(x = 3\) and \(x = 12\).
Step by step solution
01
Understand the Problem
We are given the revenue function \(R(x) = -x^2 + 30x\) and the cost function \(C(x) = 15x + 36\). We need to determine the value of \(x\) that maximizes the profit and also find the break-even points where the profit is zero.
02
Define the Profit Function
The profit function \(P(x)\) is defined as the difference between the revenue and the cost. Therefore, \(P(x) = R(x) - C(x) = (-x^2 + 30x) - (15x + 36)\). Simplifying this expression gives the profit function: \(P(x) = -x^2 + 15x - 36\).
03
Find the Maximum Profit
The function \(P(x) = -x^2 + 15x - 36\) is a quadratic function that opens downwards (as the coefficient of \(x^2\) is negative). The maximum profit occurs at the vertex of this parabola, which is given by \(x = \frac{-b}{2a}\). In \(P(x)\), \(a = -1\) and \(b = 15\), so \(x = \frac{-15}{2(-1)} = 7.5\). Thus, the value of \(x\) that maximizes profit is \(x = 7.5\).
04
Calculate the Break-even Points
To find the break-even points, we set the profit function equal to zero: \(-x^2 + 15x - 36 = 0\). This is a standard quadratic equation. Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = -1\), \(b = 15\), \(c = -36\), we calculate the discriminant \(b^2 - 4ac = 15^2 - 4(-1)(-36) = 225 - 144 = 81\). The solutions are \(x = \frac{-15 \pm \sqrt{81}}{-2}\), which simplifies to \(x = 3\) and \(x = 12\). Thus, the break-even quantities are \(x = 3\) and \(x = 12\).
05
Graph the Solution
To visualize the solution, we need to graph \(P(x) = -x^2 + 15x - 36\). The profit function is a downward-opening parabola with a vertex at \((7.5, P(7.5))\), and it intersects the x-axis at \(x = 3\) and \(x = 12\). These points highlight the maximum profit and break-even quantities, respectively.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Function
A quadratic function is a type of polynomial function with the general form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In the context of profit optimization, quadratic functions are crucial because they can model the relationship between business metrics such as revenue, cost, and profit, depending on the units sold or produced.
Quadratic functions produce a parabola when graphed. The parabola can either open upwards or downwards.
For profit scenarios, the quadratic nature helps identify key points such as maximum profit and break-even points, offering significant insights into business strategy. The vertex of the parabola gives either the maximum or minimum value, depending on the direction of the parabola.
Quadratic functions produce a parabola when graphed. The parabola can either open upwards or downwards.
- If the coefficient \( a \) is positive, the parabola opens upwards.
- If \( a \) is negative, the parabola opens downwards, indicating that there is a maximum point鈥攙ery important for maximizing profit!
For profit scenarios, the quadratic nature helps identify key points such as maximum profit and break-even points, offering significant insights into business strategy. The vertex of the parabola gives either the maximum or minimum value, depending on the direction of the parabola.
Revenue and Cost Functions
Revenue and cost functions are essential in understanding a business's financial performance. The revenue function \( R(x) \) represents the money made from selling \( x \) units of a product or service. It often involves variables and can be modeled with linear or quadratic expressions.
On the other hand, the cost function \( C(x) \) accounts for the total expenses incurred in producing \( x \) units. Like revenue functions, they can be linear or quadratic, factoring in fixed costs and variable production costs.
In this exercise, the revenue function \( R(x) = -x^2 + 30x \) is quadratic, implying a complex relationship with the number of goods produced. This can account for diminishing returns or market saturation. Meanwhile, the cost function \( C(x) = 15x + 36 \) is linear. To find the profit, subtract the cost from the revenue: \( P(x) = R(x) - C(x) \). This new function, \( P(x) \), is again quadratic, essential for finding critical points like maximum profit and break-even levels.
On the other hand, the cost function \( C(x) \) accounts for the total expenses incurred in producing \( x \) units. Like revenue functions, they can be linear or quadratic, factoring in fixed costs and variable production costs.
- The revenue function is often expressed as \( R(x) = p \cdot x \), where \( p \) is the price per unit.
- The cost function might be \( C(x) = cx + F \), with \( c \) being variable cost per unit and \( F \) the fixed cost.
In this exercise, the revenue function \( R(x) = -x^2 + 30x \) is quadratic, implying a complex relationship with the number of goods produced. This can account for diminishing returns or market saturation. Meanwhile, the cost function \( C(x) = 15x + 36 \) is linear. To find the profit, subtract the cost from the revenue: \( P(x) = R(x) - C(x) \). This new function, \( P(x) \), is again quadratic, essential for finding critical points like maximum profit and break-even levels.
Vertex of a Parabola
The vertex of a parabola is a key concept in quadratic functions, especially in profit optimization. For a quadratic function \( ax^2 + bx + c \), the vertex is the point \((h, k)\), where the function reaches its maximum or minimum value.
For a parabola described by \( P(x) = ax^2 + bx + c \), the \( x \)-coordinate of the vertex \( h \) is given by the formula \( h = \frac{-b}{2a} \). This is crucial in profit optimization as this point represents the quantity of goods that maximizes (or minimizes) profit, depending on the orientation of the parabola.
Given the profit function \( P(x) = -x^2 + 15x - 36 \), we identify \( a = -1 \) and \( b = 15 \). Calculating the vertex gives us \( x = \frac{-15}{2(-1)} = 7.5 \).
This simple formula for the vertex is an invaluable tool for businesses to determine optimal production levels.
For a parabola described by \( P(x) = ax^2 + bx + c \), the \( x \)-coordinate of the vertex \( h \) is given by the formula \( h = \frac{-b}{2a} \). This is crucial in profit optimization as this point represents the quantity of goods that maximizes (or minimizes) profit, depending on the orientation of the parabola.
Given the profit function \( P(x) = -x^2 + 15x - 36 \), we identify \( a = -1 \) and \( b = 15 \). Calculating the vertex gives us \( x = \frac{-15}{2(-1)} = 7.5 \).
- Here, the vertex indicates the maximum profit point, since the parabola opens downwards (\( a \) is negative).
- The "y" value of the vertex \( P(7.5) \) can be calculated to determine the maximum profit itself.
This simple formula for the vertex is an invaluable tool for businesses to determine optimal production levels.
