# Venn-Diagram

Problem#07 (Chapter 7.2): Venn-Diagram [6] A survey of 1200 people in a town in DC indicates that 840 people own microwave ovens, 740 people own DVD players, and 590 people own microwave ovens and DVD players together respectively. Design an appropriate Venn-Diagram and answer the following questions: (a) How many people in the survey own either a microwave oven or a DVD player? (b) How many people own neither a microwave oven nor a DVD player? (c) How many people own a microwave oven and do not own a DVD player? Problem#08 (Chapter 7.2): Venn-Diagram Construction Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9), A = {3, 6, 9) and B = {3, 4, 5, 6, 7). (a)Plug in the corresponding elements in the given sets A and B in the Venn diagram as shown below, and fill in the gaps indicating the appropriate set operations for the shaded regions. U B A B B o (i) A_B (ii) A_B (iii) A- (b) Answer the following questions using the Venn-Diagram constructed in part-(a): (1) AB = (ii) AUB= (iii) A’ = (iv) B= (v) (ANB)’ = (vi) (A U B)’= (vii) n(U)= Problem#04 [Chapter #5.1]: Mixed Inequality Problem. [6] Graph the following solution set, and determine the feasible region in graph. Also determine corresponding control points of the bounded region. y x; ys (16-x?); and y2 (-25 + x); Problem#05 [Chapter 4.1]: System of Linear Functions [6] A small plant manufactures riding lawn mowers. The plant has a net fixed cost (leases, insurance, and so on) of $56000/day and variable costs (labor, materials, and so on) of $1600 per unit produced. The mowers are sold for $2000 each unit per day. Write down the appropriate cost and revenue equations for the problem respectively. Using these equations, answer the following questions: (a) How many units must be manufactured and sold each day for the company to maintain break-even trade? (b) Plot both equations simultaneously in a graph, and clearly indicate break-even point, equilibrium point and equilibrium price in this trade to maintain revenue as expected. Interpret the significance of the regions to the left- and right-side of the break-even point in this business. Problem# 06 [Chapter 5.3]: Linear Programming Application [6] A fruit grower can use two types of fertilizers in his orange grove, Brand-A and Brand-B. Each brand of bag is composed of different types of fertilizers as mentioned below: (a) Brand-A contains 2 pounds of Nitrogen, 8 pounds of Phosphorous, and 2 pounds of Chlorine in each bag. (b) Brand-B contains 9 pounds of Nitrogen, 8 pounds of Phosphorous, and 1 pound of Chlorine in each bag. Test indicates that the grove needs at least 2000 pounds of phosphorous and at most 400 pounds of chlorine to maintain the quality of fruits produced in the grove. Analyze the corresponding system of inequalities by taking into account of these conditions in appropriate way in a graph, and also solve it properly to address the following cases: (i) If the grower wants to maximize the amount of nitrogen added to the grove, how many bags of each mixes should be used, and how much nitrogen would be added? (ii) Alternatively, if he wants to minimize the amount of nitrogen added to the grove, how many bags of each mixes should be used, and how much nitrogen would be added? Problem#01 [Chapter 4.1]: System of Linear Equations [6] Express the given system of linear equations in an augmented matrix, and transform the matrix into the RREF form. x1 – 3×2 = -1; 3×1 – x3 = 5; and x2 – 2×3 = -1; Verify your final solution. Problem#02 [Chapter 4.5): Inverse of a Square Matrix. [6] Using any method of your choice, determine the inverse of a matrix [AL] for the given square matrix [A] as shown below: [7 2 3 5 Also verify your result by proving that [A4] [A]= I. Problem#03 [Chapter 4.1]: System of Linear Equations. [6] Animals in an experiment are to be kept under a strict diet. Each animal should receive 45 grams of protein and 7.0 grams of fat. The lab-technician is able to purchase two different types of food mixes: MIX’A’ and MIX’B’ with the contents as follow. (i) MIX’A’ contains 30% protein and 8% fat, and (ii) MIX’B’ contains 40% protein and 4% fat respectively. Solve the following questions: (a) Write down the preliminary equations for protein and fat respectively, to fulfill the dieting restrictions of the animal. (b) How many grams of each mixes should be used to obtain the right diet for each animal based on the restrictions? (c) Assume that you decided to prepare a new mixture, MIX’C’ using your choice of proportions of the mixes: MIX’A’ and MIX’B’ such as mixing 40% of MIX’A’ and 60% of MIX’B’ to prepare a 100% homogeneous solution for a new diet with no-restrictions. Using the preliminary equations obtained in part (a) for the restricted diet and proportionality relation of the new MIX ‘C’, determine two independent linear equations for the protein as a function of the concentration of MIX’A’ and MIX’B’ respectively. Simultaneously plot the both graphs for the protein using the corresponding equations. Eastern UTC-04:00 UTC-05:00 Problem#01 [Chapter 3.3]: Financial Mathematics [10] A company estimates that it will need $199,000 in 5-years to replace the existing computer networking system in its current business. If it establishes a sinking fund by making fixed monthly payments into an account paying 3.1% interest compounded monthly, how much should each payment be? Problem# 02 [Chapter 5.3]: Linear Objective Function and Optimization Application [10] On a special occasion, renowned electronic giant retailer Best Buy sold a variety of TVs in discounted price or on huge sale. Perform the following Tasks (1-4) systematically to analyze the net profit or loss made by the retailer in this event: Task-1: Write down an appropriate equation for the net profit function, P in order to account for the net profit made by Best Buy after selling (1) x-number of televisions (TVS) that made $250 profit on sale of each SONY TVs, and (ii) y-number of televisions that made $10,500 profit on sale of each PHILIP TVs respectively, in that event. Task-2: Profit margin, P as discussed in Task-1 was subject to the following constraints applied on the products. a. Maximum cost price was $5000 to manufacture each piece of SONY TVs, and $10,000 to manufacture each piece of PHILIP TVs respectively. All TVs were manufactured at a maximum cost of $10 x10 as given in Plot No 1: Plot No. 1: ($5000 x + $ 10000y) < $10 million; Graph the Plot No. 1 up to the scale in a paper. b. 10 and 150 employees worked to assemble x-number of SONY TVs and y-number of PHILIP TVs respectively, for a maximum of 46000 working hours as expressed below in Plot No. 2: Plot No. 2: (10x+150y) 46000; Graph the Plot No. 2 up to the scale in the same paper. c. Relative demand of the SONY TVs and PHILIP TVs in the market was found to be in proportion as given below: Plot No. 3: y = (x+10); Graph the Plot No. 3 up to the scale in the same paper. Task-3: Determine the optimal quantities of SONY TVS (Xo) and Philip TVs (yo) needed to be sold to maximize the profit (P) in this event that was subject to the constraints as mentioned above in Task-2. Task-4: What is the maximum profit (Pmax) that Best Buy could have made by selling X, SONY TVs and yo PHILIP TVS? Problem#04 (Chapter 7.2): Venn-Diagram and Counting [10] A survey of 1200 people in a town in DC indicates that 840 people own microwave ovens, 740 people own DVD players, and 590 people own microwave ovens and DVD players together respectively. Design an appropriate Venn-Diagram and answer the following questions: (a) How many people in the survey own either a microwave oven or a DVD player? (b) How many people own neither a microwave oven nor a DVD player? (c) How many people own a microwave oven and do not own a DVD player? Problem#01 [Chapter 3.3]: Financial Mathematics [10] A company estimates that it will need $199,000 in 5-years to replace the existing computer networking system in its current business. If it establishes a sinking fund by making fixed monthly payments into an account paying 3.1% interest compounded monthly, how much should each payment be? Problem# 02 [Chapter 5.3]: Linear Objective Function and Optimization Application [10] On a special occasion, renowned electronic giant retailer Best Buy sold a variety of TVs in discounted price or on huge sale. Perform the following Tasks (1-4) systematically to analyze the net profit or loss made by the retailer in this event: Task-1: Write down an appropriate equation for the net profit function, P in order to account for the net profit made by Best Buy after selling (1) x-number of televisions (TVS) that made $250 profit on sale of each SONY TVs, and (ii) y-number of televisions that made $10,500 profit on sale of each PHILIP TVs respectively, in that event. Task-2: Profit margin, P as discussed in Task-1 was subject to the following constraints applied on the products. a. Maximum cost price was $5000 to manufacture each piece of SONY TVs, and $10,000 to manufacture each piece of PHILIP TVs respectively. All TVs were manufactured at a maximum cost of $10 x10 as given in Plot No 1: Plot No. 1: ($5000 x + $ 10000y) < $10 million; Graph the Plot No. 1 up to the scale in a paper. b. 10 and 150 employees worked to assemble x-number of SONY TVs and y-number of PHILIP TVs respectively, for a maximum of 46000 working hours as expressed below in Plot No. 2: Plot No. 2: (10x+150y) 46000; Graph the Plot No. 2 up to the scale in the same paper. c. Relative demand of the SONY TVs and PHILIP TVs in the market was found to be in proportion as given below: Plot No. 3: y = (x+10); Graph the Plot No. 3 up to the scale in the same paper. Task-3: Determine the optimal quantities of SONY TVS (Xo) and Philip TVs (yo) needed to be sold to maximize the profit (P) in this event that was subject to the constraints as mentioned above in Task-2. Task-4: What is the maximum profit (Pmax) that Best Buy could have made by selling X, SONY TVs and yo PHILIP TVS?

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