CONCORDIA UNIVERSITY MATH 209 – APRIL 2024 FINAL EXAM SIMILAR QUESTIONS

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CONCORDIA UNIVERSITY MATH EXAM

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QUESTION.1 

A company manufactures and sells 𝑥 items per week. The weekly cost function and price-demand equations are, respectively,

C ( x ) = 60 x + 20000 p = 120 − 1 32 x

where 𝑥 is the weekly demand and 𝑝 is the price in dollars. Assume that the price and demand are both positive. The government decides to tax the company $2 for each item it produces.

Find the production level and price per item that produces the maximum profit.

What is the maximum profit?

Solution

In order to find the production level and price per item that produce the maximum profit, we need to find the profit equation, 𝑃(𝑥).

We will use the equation that relates Profit with Revenue and Cost:

Profit = R e v e n u e − C o s t P ( x ) = R ( x ) − C ( x )

The Cost function is already given in the question, however, we need to consider the fact that the government is adding a $2 tax on each item produced. This means we need to add 2𝑥 to the cost function:

C ( x ) = 60 x + 20000 + 2 x

The new cost function then becomes:

C ( x ) = 62 x + 20000

To find the Revenue function, we need to multiply the price by the quantity of items sold. This is represented by:

R ( x ) = p ⋅ x

Where 𝑝 is the price equation given in the question and 𝑥 is the quantity sold.

R ( x ) = ( 120 − 1 32 x ) ⋅ x R ( x ) = 120 x − 1 32 x 2

Now that we have the Revenue function, we can substitute it into the Profit function and get a Profit function in terms of 𝑥.

P ( x ) = R ( x ) − C ( x ) P ( x ) = ( 120 x − 1 32 x 2 ) − ( 62 x + 20000 ) P ( x ) = 120 x − 1 32 x 2 − 62 x − 20000 F ( x ) = − 1 32 x 2 + 58 x − 20000

The profit function obtained above is a quadratic function that opens downwards, so it has a vertex that is a maximum. (See graph below)

To find the production level and the price that produce the maximum profit, we need to find the 𝑥 value of the vertex. There are two methods to find the vertex of a quadratic function:

From the calculations above, we can see that the production level – the number of items that need to be produced and sold to maximize the profit – is 𝑥 = 928.

Price per item:

The price of each item can be calculated using the price function given in the question:

p = 120 − 1 32 x Substitute x = 928 in for  x . p = 120 − 1 32 ( 928 ) p = 120 − 1 32 ( 928 ) p = 120 − 29 p = 91

So the price that maximizes the profit is $91.

To find the maximum monthly profit, we substitute 𝑥 = 928 into the Profit and calculate this maximum value:

P ( x ) = − 1 32 x 2 + 58 x − 20000 P ( 928 ) = − 1 32 ( 928 ) 2 + 58 ( 928 ) − 20000 P ( 928 ) = 6912

So the maximum monthly profit is $6912.

Final Answer:

The production level that maximizes profit is 928 items per week, with a corresponding price of $91 per item. The maximum profit is $6912 per week.

QUESTION.2

(8 points) A company manufactures and sells 𝑥 items per month. The monthly cost function and price-demand equation are, respectively,

C ( x ) = 50 x + 35000 p = 150 − x 30

where 𝑥 is the monthly demand and 𝑝 is the price in dollars. Assume that the price and demand are both positive. The government decides to tax the company $8 for each item it produces.

Find the production level and price per item that produce the maximum profit.

What is the maximum monthly profit?

Solution

In order to find the production level and price per item that produce the maximum profit, we need to find the profit equation, 𝑃(𝑥).

We will use the equation that relates Profit with Revenue and Cost:

Profit = R e v e n u e − C o s t P ( x ) = R ( x ) − C ( x )

The Cost function is already given in the question, however, we need to consider the fact that the government is adding a $8 tax on each item produced. This means we need to add 8𝑥 to the cost function:

C ( x ) = 50 x + 35000 + 8 x

The new cost function then becomes:

C ( x ) = 58 x + 35000

To find the Revenue function, we need to multiply the price by the quantity of items sold. This is represented by:

R ( x ) = p ⋅ x

Where 𝑝 is the price equation given in the question and 𝑥 is the quantity sold.

R ( x ) = ( 150 − x 30 ) ⋅ x R ( x ) = 150 x − x 2 30

Now that we have the Revenue function, we can substitute it into the Profit function and get a Profit function in terms of 𝑥.

P ( x ) = R ( x ) − C ( x ) P ( x ) = ( 150 x − x 2 30 ) − ( 58 x + 35000 ) P ( x ) = 150 x − x 2 30 − 58 x − 35000 P ( x ) = − x 2 30 + 92 x − 35000

The profit function obtained above is a quadratic function that opens downwards, so it has a vertex that is a maximum. (See graph below)

To find the production level and the price that produce the maximum profit, we need to find the 𝑥 value of the vertex. There are two methods to find the vertex of a quadratic function:

From the calculations above, we can see that the production level – the number of items that need to be produced and sold to maximize the profit – is 𝑥 = 1380.

Price per item:

The price of each item can be calculated using the price function given in the question:

p = 150 − x 30 Substitute x = 1380 in for  x . p = 150 − 1 30 ( 1380 ) p = 150 − 46 p = 104

So the price that maximizes the profit is $104.

To find the maximum monthly profit, we substitute 𝑥 = 1380 into the Profit and calculate this maximum value:

P ( x ) = − 1 30 x 2 + 92 x − 35000 P ( 928 ) = − 1 30 ( 1380 ) 2 + 92 ( 1380 ) − 35000 P ( 1380 ) = 28480

So the maximum monthly profit is $28480.

Final Answer:

The production level that maximizes profit is 1380 items per week, with a corresponding price of $104 per item. The maximum profit is $28480 per week.

QUESTION.3

(8 points) A factory produces and sells 𝑥 gadgets per month. The monthly cost function and price-demand equation are, respectively,

C ( x ) = 80 x + 50000 p = 250 − x 50

where 𝑥 is the monthly demand and 𝑝 is the price in dollars. Assume that the price and demand are both positive. The government imposes a $12 tax for each item produced.

Find the production level and price per item that produce the maximum profit.

What is the maximum monthly profit?

Solution

In order to find the production level and price per item that produce the maximum profit, we need to find the profit equation, 𝑃(𝑥).

We will use the equation that relates Profit with Revenue and Cost:

Profit = R e v e n u e − C o s t P ( x ) = R ( x ) − C ( x )

The Cost function is already given in the question, however, we need to consider the fact that the government is adding a $12 tax on each item produced. This means we need to add 12𝑥 to the cost function:

C ( x ) = 80 x + 50000 + 12 x

The new cost function then becomes:

C ( x ) = 92 x + 50000

To find the Revenue function, we need to multiply the price by the quantity of items sold. This is represented by:

R ( x ) = p ⋅ x

Where 𝑝 is the price equation given in the question and 𝑥 is the quantity sold.

R ( x ) = ( 250 − x 50 ) ⋅ x R ( x ) = 250 x − x 2 50

Now that we have the Revenue function, we can substitute it into the Profit function and get a Profit function in terms of 𝑥.

P ( x ) = R ( x ) − C ( x ) P ( x ) = ( 250 x − x 2 50 ) − ( 92 x + 50000 ) P ( x ) = 250 x − x 2 50 − 92 x − 50000 P ( x ) = − x 2 50 + 158 x − 50000

The profit function obtained above is a quadratic function that opens downwards, so it has a vertex that is a maximum. (See graph below)

To find the production level and the price that produce the maximum profit, we need to find the 𝑥 value of the vertex. There are two methods to find the vertex of a quadratic function:

From the calculations above, we can see that the production level – the number of items that need to be produced and sold to maximize the profit – is 𝑥 = 3950.

Price per item:

The price of each item can be calculated using the price function given in the question:

p = 250 − x 50 Substitute x = 3950 in for  x . p = 250 − 3950 50 p = 250 − 79 p = 171

So the price that maximizes the profit is $171.

To find the maximum monthly profit, we substitute 𝑥 = 1380 into the Profit and calculate this maximum value:

P ( x ) = − x 2 50 + 158 x − 50000 P ( 928 ) = − ( 3950 ) 2 50 + 158 ( 3950 ) − 50000 P ( 3950 ) = 262050

So the maximum monthly profit is $262050.

Final Answer:

The production level that maximizes profit is 3950 items per week, with a corresponding price of $171 per item. The maximum profit is $262050 per week.

QUESTION.4

(8 points) A factory produces and sells 𝑥 gadgets per month. The monthly cost function and price-demand equation are, respectively,

C ( x ) = 45 x + 40000 p = 175 − x 36

where 𝑥 is the monthly demand and 𝑝 is the price in dollars. Assume that the price and demand are both positive. The government imposes a $15 tax for each item produced.

Find the production level and price per item that produce the maximum profit.

What is the maximum monthly profit?

Solution

In order to find the production level and price per item that produce the maximum profit, we need to find the profit equation, 𝑃(𝑥).

We will use the equation that relates Profit with Revenue and Cost:

Profit = R e v e n u e − C o s t P ( x ) = R ( x ) − C ( x )

The Cost function is already given in the question, however, we need to consider the fact that the government is adding a $12 tax on each item produced. This means we need to add 12𝑥 to the cost function:

C ( x ) = 45 x + 40000 + 15 x

The new cost function then becomes:

C ( x ) = 60 x + 40000

To find the Revenue function, we need to multiply the price by the quantity of items sold. This is represented by:

R ( x ) = p ⋅ x

Where 𝑝 is the price equation given in the question and 𝑥 is the quantity sold.

R ( x ) = ( 175 − x 36 ) ⋅ x R ( x ) = 175 x − x 2 36

Now that we have the Revenue function, we can substitute it into the Profit function and get a Profit function in terms of 𝑥.

P ( x ) = R ( x ) − C ( x ) P ( x ) = ( 175 x − x 2 36 ) − ( 60 x + 40000 ) P ( x ) = 175 x − x 2 36 − 60 x − 40000 P ( x ) = − x 2 36 + 115 x − 40000

The profit function obtained above is a quadratic function that opens downwards, so it has a vertex that is a maximum. (See graph below)

To find the production level and the price that produce the maximum profit, we need to find the 𝑥 value of the vertex. There are two methods to find the vertex of a quadratic function:

From the calculations above, we can see that the production level – the number of items that need to be produced and sold to maximize the profit – is 𝑥 = 2070.

Price per item:

The price of each item can be calculated using the price function given in the question:

p = 175 − x 36 Substitute x = 2070 in for  x . p = 175 − 2070 36 p = 175 − 57.5 p = 117.50

So the price that maximizes the profit is $117.50.

To find the maximum monthly profit, we substitute 𝑥 = 2070 into the Profit and calculate this maximum value:

P ( x ) = − x 2 36 + 115 x − 40000 P ( 2070 ) = − ( 2070 ) 2 36 + 115 ( 2070 ) − 40000 P ( 2070 ) = 79025

So the maximum monthly profit is $79025.

Final Answer:

The production level that maximizes profit is 2070 items per week, with a corresponding price of $117.50 per item. The maximum profit is $79025 per week.

QUESTION.5

(8 points) A factory produces and sells 𝑥 gadgets per month. The monthly cost function and price-demand equation are, respectively,

C ( x ) = 80 x + 48000 p = 220 − 2x 45

where 𝑥 is the monthly demand and 𝑝 is the price in dollars. Assume that the price and demand are both positive. The government imposes a $8 tax for each item produced.

Find the production level and price per item that produce the maximum profit.

What is the maximum monthly profit?

Solution

In order to find the production level and price per item that produce the maximum profit, we need to find the profit equation, 𝑃(𝑥).

We will use the equation that relates Profit with Revenue and Cost:

Profit = R e v e n u e − C o s t P ( x ) = R ( x ) − C ( x )

The Cost function is already given in the question, however, we need to consider the fact that the government is adding a $8 tax on each item produced. This means we need to add 8𝑥 to the cost function:

C ( x ) = 80 x + 48000 + 8 x

The new cost function then becomes:

C ( x ) = 88 x + 48000

To find the Revenue function, we need to multiply the price by the quantity of items sold. This is represented by:

R ( x ) = p ⋅ x

Where 𝑝 is the price equation given in the question and 𝑥 is the quantity sold.

R ( x ) = ( 220 − 2 x 45 ) ⋅ x R ( x ) = 220 x − 2 x 2 45

Now that we have the Revenue function, we can substitute it into the Profit function and get a Profit function in terms of 𝑥.

P ( x ) = R ( x ) − C ( x ) P ( x ) = ( 220 x − 2 x 2 45 ) − ( 88 x + 48000 ) P ( x ) = 220 x − 2 x 2 45 − 88 x − 48000 P ( x ) = − 2 x 2 45 + 132 x − 48000

The profit function obtained above is a quadratic function that opens downwards, so it has a vertex that is a maximum. (See graph below)

To find the production level and the price that produce the maximum profit, we need to find the 𝑥 value of the vertex. There are two methods to find the vertex of a quadratic function:

From the calculations above, we can see that the production level – the number of items that need to be produced and sold to maximize the profit – is 𝑥 = 1485.

Price per item:

The price of each item can be calculated using the price function given in the question:

p = 220 − 2 x 45 Substitute x = 1485 in for  x . p = 220 − 2 ( 1485 ) 45 p = 220 − 66 p = 154

So the price that maximizes the profit is $154.

To find the maximum monthly profit, we substitute 𝑥 = 2070 into the Profit and calculate this maximum value:

P ( x ) = − 2 x 2 45 + 132 x − 48000 P ( 2070 ) = − 2 ( 1485 ) 2 45 + 132 ( 1485 ) − 48000 P ( 1485 ) = 50010

So the maximum monthly profit is $50010

Final Answer:

The production level that maximizes profit is 1485 items per week, with a corresponding price of $154 per item. The maximum profit is $50010 per week

QUESTION.6

(8 points) A factory produces and sells 𝑥 gadgets per month. The monthly cost function and price-demand equation are, respectively,

C ( x ) = 55 x + 37000 p = 180 − x 25

where 𝑥 is the monthly demand and 𝑝 is the price in dollars. Assume that the price and demand are both positive. The government imposes a $11 tax for each item produced.

Find the production level and price per item that produce the maximum profit.

What is the maximum monthly profit?

Solution

In order to find the production level and price per item that produce the maximum profit, we need to find the profit equation, 𝑃(𝑥).

We will use the equation that relates Profit with Revenue and Cost:

Profit = R e v e n u e − C o s t P ( x ) = R ( x ) − C ( x )

The Cost function is already given in the question, however, we need to consider the fact that the government is adding a $8 tax on each item produced. This means we need to add 8𝑥 to the cost function:

C ( x ) = 55 x + 37000 + 11 x

The new cost function then becomes:

C ( x ) = 66 x + 37000

To find the Revenue function, we need to multiply the price by the quantity of items sold. This is represented by:

R ( x ) = p ⋅ x

Where 𝑝 is the price equation given in the question and 𝑥 is the quantity sold.

R ( x ) = ( 180 − x 25 ) ⋅ x R ( x ) = 180 x − x 2 25

Now that we have the Revenue function, we can substitute it into the Profit function and get a Profit function in terms of 𝑥.

P ( x ) = R ( x ) − C ( x ) P ( x ) = ( 180 x − x 2 25 ) − ( 66 x + 37000 ) P ( x ) = 180 x − x 2 25 − 66 x − 37000 . P ( x ) = − x 2 25 + 114 x − 37000

The profit function obtained above is a quadratic function that opens downwards, so it has a vertex that is a maximum. (See graph below)

CONCORDIA UNIVERSITY MATH EXAM

To find the production level and the price that produce the maximum profit, we need to find the 𝑥 value of the vertex. There are two methods to find the vertex of a quadratic function:

From the calculations above, we can see that the production level – the number of items that need to be produced and sold to maximize the profit – is 𝑥 = 1425.

Price per item:

The price of each item can be calculated using the price function given in the question:

p = 180 − x 25 Substitute x = 1485 in for  x . p = 180 − 1425 25 p = 180 − 57 p = 123

So the price that maximizes the profit is $123.

To find the maximum monthly profit, we substitute 𝑥 = 2070 into the Profit and calculate this maximum value:

P ( x ) = − x 2 25 + 114 x − 37000 P ( 2070 ) = − ( 1425 ) 2 25 + 114 ( 1425 ) − 37000 P ( 1425 ) = 44225

So the maximum monthly profit is $44225

Final Answer:

The production level that maximizes profit is 1425 items per week, with a corresponding price of $123 per item. The maximum profit is $44225 per week.

CONCORDIA UNIVERSITY MATH EXAM

Go to the original Question: 

Concordia University Math Exam, School of Graduate Studies

Concordia University offers comprehensive exam guides to help students prepare for their exams effectively. Here’s a detailed overview of what to expect in a typical exam guide at Concordia:

1. Exam Schedule and Format

  • Exam Dates and Times: The exam guide includes a detailed schedule with specific dates, times, and locations for each exam. Students should verify their exam schedules on the official Concordia University portal, as changes can occur.
  • Format: The guide will specify whether the exam is in-person, online, or take-home. It will also outline the structure of the exam, such as multiple-choice, short-answer, essay, problem-solving, or a combination.

2. Course Content and Study Resources

  • Topics Covered: A breakdown of the topics and chapters that will be covered in the exam is provided. This helps students focus their revision on the most important content.
  • Key Concepts: Important theories, models, formulas, or concepts that are crucial for the exam are highlighted.
  • Study Tips: Recommendations for textbooks, lecture notes, and additional resources are provided. There may also be tips on time management and study techniques.

3. Permitted Materials

  • Allowed Items: The guide will specify what materials students are allowed to bring into the exam, such as calculators, formula sheets, notes, or textbooks (for open-book exams).
  • Prohibited Items: A list of items not allowed in the exam room, such as electronic devices, is also provided to ensure compliance with exam regulations.

4. Academic Integrity

  • Plagiarism and Cheating Policies: Concordia’s exam guide emphasizes the importance of academic integrity, outlining the consequences of plagiarism, cheating, and other forms of academic misconduct.
  • Ethical Guidelines: It provides guidelines on maintaining honesty and fairness during exams, whether in-person or online.

5. Special Accommodations

  • Accessibility Services: Information on how students with disabilities can request special accommodations for their exams. This might include extended time, separate rooms, or other necessary adjustments.
  • Application Procedures: The process for applying for accommodations, including deadlines and required documentation, is detailed.

6. Exam Preparation Workshops

  • Workshops and Sessions: Details of any workshops, review sessions, or study groups organized by the university to help students prepare for their exams.
  • Availability: Information on how to sign up and the topics that will be covered in these sessions.

7. Exam Day Instructions

  • Arrival Time: Guidelines on when to arrive at the exam venue, considering possible delays and the time needed to find the correct location.
  • Identification: The type of identification required to enter the exam room, usually a student ID card.
  • Exam Protocols: Instructions on what to do before, during, and after the exam, such as filling out answer sheets, managing time effectively, and when to leave the exam room.

8. Post-Exam Information

  • Results Release: Information on when and how exam results will be released.
  • Appeals Process: Steps to take if a student wishes to appeal their exam grade or request a re-evaluation.
  • Feedback: How to obtain feedback on exam performance if available.

9. Contact Information

  • Support Services: Contact details for academic advisors, professors, or the exam office in case students have any questions or need further clarification.
  • Technical Support: For online exams, information on how to reach IT support in case of technical difficulties during the exam.

10. Frequently Asked Questions (FAQs)

  • Common Queries: The guide often includes a section addressing common questions about exams, such as what to do if a student is sick on the day of the exam or how to handle conflicts in the exam schedule.

This detailed exam guide is typically accessible through Concordia University’s student portal or provided by course instructors. It is essential for students to review the guide thoroughly to ensure they are well-prepared and compliant with all exam regulations.

Detailed Concordia University Math Exam Guide

Preparing for a Concordia University Math Exam requires a strategic approach to ensure success. The Concordia University Math Exam guide is an essential resource that provides students with detailed information on what to expect during their exams. Whether you are a first-year student or nearing graduation, understanding the structure and content of a Concordia University Math Exam is crucial for achieving high marks.

Exam Structure and Format

The Concordia University Math Exam typically varies in structure depending on the course level. For introductory courses, a Concordia University Math Exam might focus more on fundamental concepts such as algebra and basic calculus. In contrast, advanced-level Concordia University Math Exams may cover more complex topics like differential equations, linear algebra, or advanced statistical methods. Regardless of the course level, each Concordia University Math Exam will generally consist of a mix of multiple-choice questions, short-answer problems, and long-form problem-solving tasks. Knowing the exact format of your specific Concordia University Math Exam can help you tailor your study sessions accordingly.

Topics Covered

The Concordia University Math Exam guide provides a detailed outline of the topics that will be covered in the exam. For example, a Concordia University Math Exam in calculus might include topics such as limits, derivatives, integrals, and their applications. In contrast, a Concordia University Math Exam in linear algebra would focus on matrices, vectors, and systems of linear equations. The guide ensures that students are aware of the key areas to focus on, making it easier to organize and prioritize their study time.

Study Resources and Tips

When preparing for a Concordia University Math Exam, it is essential to use the right resources. The Concordia University Math Exam guide usually suggests textbooks, online resources, and previous exam papers as valuable study aids. Practice problems, especially those found in past Concordia University Math Exams, are particularly useful for getting a feel for the types of questions that might appear. Additionally, the guide may offer study tips such as time management strategies, how to tackle difficult problems, and techniques for retaining complex information, all of which are crucial for succeeding in a Concordia University Math Exam.

Permitted Materials

Understanding what materials are allowed during the Concordia University Math Exam is vital. The guide clearly states whether students can use calculators, formula sheets, or any other reference materials during their Concordia University Math Exam. For some Concordia University Math Exams, only non-programmable calculators may be allowed, while others might permit open-book formats where textbooks and notes are accessible. Knowing the allowed materials well in advance of your Concordia University Math Exam helps prevent any last-minute surprises.

Academic Integrity

Maintaining academic integrity during a Concordia University Math Exam is paramount. The exam guide emphasizes the importance of honesty and fairness, outlining the severe consequences of any form of cheating or plagiarism in a Concordia University Math Exam. This includes using unauthorized materials, copying from peers, or any other unethical behavior. The guide often provides scenarios that could lead to academic misconduct during a Concordia University Math Exam and advises students on how to avoid these pitfalls.

Special Accommodations

For students who require special accommodations during a Concordia University Math Exam, the guide provides detailed information on how to apply for these adjustments. This might include additional time, separate exam rooms, or assistive technologies. The Concordia University Math Exam guide outlines the process for requesting accommodations, including deadlines and the necessary documentation, ensuring that all students have an equal opportunity to succeed.

Exam Day Instructions

On the day of the Concordia University Math Exam, it’s important to follow the instructions provided in the exam guide. Students should arrive early to their Concordia University Math Exam location, bringing with them all required identification and permitted materials. The guide also offers advice on how to manage exam anxiety and maintain focus during the Concordia University Math Exam. Understanding these instructions can make a significant difference in how smoothly the Concordia University Math Exam day goes.

Post-Exam Procedures

After the Concordia University Math Exam is completed, the guide provides information on when and how results will be released. If a student feels that there has been an error in their grading, the Concordia University Math Exam guide outlines the steps for submitting an appeal or requesting a re-evaluation. Furthermore, the guide may offer insights into how to review exam performance and what steps to take if additional support is needed for future Concordia University Math Exams.

Final Thoughts

The Concordia University Math Exam guide is an invaluable tool that helps students navigate their exam preparation with confidence. By following the guidelines and advice provided, students can enhance their understanding of the material, improve their exam-taking strategies, and ultimately perform better on their Concordia University Math Exams. Whether you are preparing for your first Concordia University Math Exam or are a seasoned student, utilizing this guide will undoubtedly contribute to your academic success.

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