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Got it! I’ll create a polished, professional assignment with a realistic teacher's name and high school, formatted like a well-structured document. I’ll include detailed explanations, diagrams, and clear formatting to make it look like a high school teacher’s assignment. I’ll also structure it for easy readability and ensure that it looks authentic. I'll generate the PDF version shortly. Stay tuned! Exploring the Dynamics of Motion and Optimization Advanced Mathematics: Calculus and Algebra Instructor: Mr. Daniel T. Reynolds Lincoln High School, Mathematics Department Assignment Overview Objective: This assignment is designed to challenge students in applying calculus and algebraic concepts to real-world situations. By solving these problems, students will gain deeper insight into mathematical modeling, optimization, rates of change, and integral applications. Instructions: • Show all steps in your solutions. • Use correct mathematical notation. • Include explanations where necessary to demonstrate understanding. • Reference the provided diagrams when applicable. • Ensure answers are accurate and properly justified. Submission Details: • Format: PDF Submission • Due Date: [Insert Due Date] • Academic Integrity: Work must be original and independently completed. Problem Set 1. Velocity and Acceleration of a Moving Object A particle moves along a straight line, and its position (in meters) at time t (in seconds) is given by: s(t) = 4t3 − 6t2 + 2t + 5 1 (a) Find the velocity function v(t). (b) Find the acceleration function a(t). (c) Determine the velocity and acceleration at t = 2 seconds. 2. Maximizing Revenue in a Business Model A company manufactures and sells x units of a product per day. The revenue function is given by: R(x) = −5x2 + 200x where R(x) represents revenue in dollars. (a) Determine the number of units that maximize revenue. (b) Calculate the maximum revenue. 3. Related Rates: Expanding Circular Oil Spill An oil spill spreads in a circular pattern. The radius of the spill increases at a rate of 0.5 m/min. (a) Derive the formula relating the area of the spill to its radius. (b) Find the rate at which the area is increasing when the radius is 10 m. 4. Tangent Line to a Function Find the equation of the tangent line to the function y = 2x3 − 5x2 + 4x − 1 at x = 3. 5. Calculating the Area Between Curves Find the area enclosed by the curves y = x2 and y = 9 by setting up and evaluating the proper integral. 6. Exponential Growth in Bacteria Population A bacterial culture grows according to the function: P (t) = 500e0.3t 2 where P (t) represents the number of bacteria after t hours. (a) Determine the population after 5 hours. (b) Find the time required for the population to reach 5000 bacteria. 7. Work Done by a Variable Force A force of F (x) = 2x3 (in Newtons) moves an object along a straight path from x = 1 to x = 4 meters. Compute the work done by integrating the force function over the given interval. 8. Solving a Logistic Growth Model A population of birds follows a logistic growth model given by P (t) = 600 1 + 9e−0.4t where P (t) represents the population at time t (in years). Find the population at t = 8 years. 9. Volume of a Solid of Revolution Find the volume of the solid formed by rotating the function y = x2 around the x-axis from x = 0 to x = 4. 10. Conic Sections: Finding the Foci of an Ellipse Determine the foci of the ellipse given by: x2 36 + y2 16 = 1. Answer Key & Solutions 1. Velocity and Acceleration (a) Velocity: v(t) = 12t2 − 12t + 2 (b) Acceleration: a(t) = 24t − 12 (c) v(2) = 26, a(2) = 36 3 2. Maximizing Revenue (a) x = 20 units maximize revenue (b) Maximum revenue: $2000 3. Related Rates (a) A = πr2 (b) dA/dt = 10π m2/min 4. Tangent Line Equation y = 49x − 124 5. Area Between Curves A = ∫ 3 −3 ( 9 − x2) dx = 36 6. Bacteria Growth (a) P (5) ≈ 933.25 (b) t ≈ 9.67 hours 7. Work Done by a Force W = ∫ 4 1 2x3dx = 127.5 J 8. Logistic Growth P (8) ≈ 447 9. Volume of Solid V = π ∫ 4 0 x4dx = 1024π 5 10. Foci of the Ellipse c = √ 36 − 16 = √ 20 ⇒ ( ± √ 20, 0 ) 4 Submission Guidelines: • Formatting: Submit as a neatly formatted PDF. • Neatness: Work should be clearly organized and legible. • Academic Integrity: Ensure work is original. Good luck! If you have any questions, please ask before the due date. I will now generate this assignment in a PDF format with proper diagrams and formatting. Stay tuned! It looks like there was an encoding issue with bullet points. I'll fix it and regenerate the PDF. Stay tuned! The assignment has been successfully formatted into a polished PDF. You can download it using the link below: Download Advanced Math Assignment Let me know if you need any modifications! 5 sandbox:/mnt/data/Advanced_Math_Assignment.pdf Exploring the Dynamics of Motion and Optimization Assignment Overview Problem Set 1. Velocity and Acceleration of a Moving Object 2. Maximizing Revenue in a Business Model 3. Related Rates: Expanding Circular Oil Spill 4. Tangent Line to a Function 5. Calculating the Area Between Curves 6. Exponential Growth in Bacteria Population 7. Work Done by a Variable Force 8. Solving a Logistic Growth Model 9. Volume of a Solid of Revolution 10. Conic Sections: Finding the Foci of an Ellipse Answer Key & Solutions 1. Velocity and Acceleration 2. Maximizing Revenue 3. Related Rates 4. Tangent Line Equation 5. Area Between Curves 6. Bacteria Growth 7. Work Done by a Force 8. Logistic Growth 9. Volume of Solid 10. Foci of the Ellipse Submission Guidelines:
