3. The cost in dollars to produce x hundreds of nails is given by C(x)=x²-12x+41. What is the minimum cost, and how many nails must be produced for minimum cost? Watch your units! quantity: C cost: 4. The profit equation, in dollars, for a business is P(x)=-4x² +400x-1000 where x is the number of cabinets sold. What is the maximum profit, and how many cabinets should be marketed and sold to get that profit? # cabinets: Max profit: 5. A profit function is given by P(x)=-x²+25x-100, where x is the number of pounds of a product sold, and profit is in dollars. a) Find the maximum profit, and the number of pounds that must be sold for maximum profit. # pounds: Max profit: b) Find the number of pounds of the product that must be sold to breakeven. or pounds
3. The cost in dollars to produce x hundreds of nails is given by C(x)=x²-12x+41. What is the minimum cost, and how many nails must be produced for minimum cost? Watch your units! quantity: C cost: 4. The profit equation, in dollars, for a business is P(x)=-4x² +400x-1000 where x is the number of cabinets sold. What is the maximum profit, and how many cabinets should be marketed and sold to get that profit? # cabinets: Max profit: 5. A profit function is given by P(x)=-x²+25x-100, where x is the number of pounds of a product sold, and profit is in dollars. a) Find the maximum profit, and the number of pounds that must be sold for maximum profit. # pounds: Max profit: b) Find the number of pounds of the product that must be sold to breakeven. or pounds
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Educational Content on Profit Functions**
This document contains exercises related to profit functions and their application in business scenarios. Here is a breakdown of the questions and solutions provided:
---
**3. Cost Function for Producing Products**
The cost to produce hundreds of nails is given by the function:
\[ C(x) = x^2 - 12x + 41 \]
where \( x \) is the number of hundreds of nails.
- **Objective:** Determine the minimum cost and how many nails must be produced to achieve this cost.
**Key Steps:**
- **Expression Analysis:** This quadratic function can be used to find the minimum point by applying vertex formulas or completing the square.
- **Results:** The minimum cost and corresponding nail quantity are not filled in but are crucial for understanding cost efficiency.
---
**4. Profit Equations for a Business**
The profit equation in dollars for a business is expressed as:
\[ P(x) = -4x^2 + 400x - 1000 \]
where \( x \) is the number of cabinets sold.
- **Objective:** Identify the maximum profit and the number of cabinets that should be marketed and sold to obtain this profit.
**Key Steps:**
- **Expression Analysis:** This is a downward-opening parabola; find the vertex to determine maximum profit.
- **Quantity & Cost:** The placeholders suggest filling in specific answers:
- Quantity: ___
- Cost: ___
- Max Profit: ___
- Cabinets (#): 50
---
**5. Profit Function Related to Product Sales**
A profit function is defined by:
\[ P(x) = x^2 + 25x - 100 \]
where \( x \) is the number of pounds of a product sold, and \( P \) is the profit in dollars.
- **Objectives:**
- **Find the maximum profit** and the number of pounds that must be sold to achieve it.
- **Determine the number of pounds required to breakeven**, where profit equals zero.
**Key Steps:**
- **Expression Analysis:** This mirrors scenarios in maximizing or minimizing quadratics by finding vertex for max profit or solving the equation for zeros to determine breakeven points.
- **Results:**
- Pounds for max: ___
- Max Profit: ___
- Pounds for breakeven: ___ or ___
---
These exercises are designed to build skills in analyzing quadratic equations as they relate](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F43bd6ecf-0a0f-4666-aaae-944e31ddf02f%2F74eba38e-6417-432c-b318-1e8b92b52b9b%2Fkssnwm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Content on Profit Functions**
This document contains exercises related to profit functions and their application in business scenarios. Here is a breakdown of the questions and solutions provided:
---
**3. Cost Function for Producing Products**
The cost to produce hundreds of nails is given by the function:
\[ C(x) = x^2 - 12x + 41 \]
where \( x \) is the number of hundreds of nails.
- **Objective:** Determine the minimum cost and how many nails must be produced to achieve this cost.
**Key Steps:**
- **Expression Analysis:** This quadratic function can be used to find the minimum point by applying vertex formulas or completing the square.
- **Results:** The minimum cost and corresponding nail quantity are not filled in but are crucial for understanding cost efficiency.
---
**4. Profit Equations for a Business**
The profit equation in dollars for a business is expressed as:
\[ P(x) = -4x^2 + 400x - 1000 \]
where \( x \) is the number of cabinets sold.
- **Objective:** Identify the maximum profit and the number of cabinets that should be marketed and sold to obtain this profit.
**Key Steps:**
- **Expression Analysis:** This is a downward-opening parabola; find the vertex to determine maximum profit.
- **Quantity & Cost:** The placeholders suggest filling in specific answers:
- Quantity: ___
- Cost: ___
- Max Profit: ___
- Cabinets (#): 50
---
**5. Profit Function Related to Product Sales**
A profit function is defined by:
\[ P(x) = x^2 + 25x - 100 \]
where \( x \) is the number of pounds of a product sold, and \( P \) is the profit in dollars.
- **Objectives:**
- **Find the maximum profit** and the number of pounds that must be sold to achieve it.
- **Determine the number of pounds required to breakeven**, where profit equals zero.
**Key Steps:**
- **Expression Analysis:** This mirrors scenarios in maximizing or minimizing quadratics by finding vertex for max profit or solving the equation for zeros to determine breakeven points.
- **Results:**
- Pounds for max: ___
- Max Profit: ___
- Pounds for breakeven: ___ or ___
---
These exercises are designed to build skills in analyzing quadratic equations as they relate
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