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1. A Venture into BusinessChapter 4

   4-1 Employees are part of the Company 4-2 Fluctuating Factors of Production 4-3 Cost, Revenue, and Profit Functions 4-4 Break-even Point 4-5 Linear Programming
2. A Venture into BusinessChapter 4 After completing this chapter, you should be able to examine managerial duties in a business prepare a payroll determine factors that influence production calculate production costs, profit, loss, revenue and break-even point write and interpret inequalities for business constraints
3. 4-1Employees are part of the Company Warm-up Word Definition Symbol/Formula Consumers Entrepreneurship Labor Management Market Economy Producers
4. 4-1Employees are part of the Company Skill 1: Payroll Gross pay = hourly rate ∙ hours worked p = rh where p = gross pay r = hourly rate h = number of hours worked Social Security tax rate on incomes under $57,600 is 6.2%. The Medicare tax rate for wages at or below $135,000 is 1.45%.
5. 4-1Employees are part of the Company Example Evelyn is paid $6.50 an hour at her business. She is single and claims 0 exemptions on her W-4 form. She worked 14 hours this week. What is Evelyn’s gross pay for the week? p = rh p = 6.5(14) p = $91
6. 4-1Employees are part of the Company Skill 2: Take-home pay T = g – (w + f) [review from chapter 2] Example What is Evelyn’s take home pay? FICA = 7.65%(91) = $6.96 = f Income tax withholding for single persons earning $91/week is $7, according to the tax withholding table. w = 7 T = g – (w + f) = 91- (7 + 6.96) = $77.04
7. 4-1Employees are part of the Company Skill 3: FICA taxes Social Security: 6.2% on incomes under $57,600 Medicare: 1.45% for all income at or below $135, 000 Example Evelyn’s father earns $68,000 a year and Hari’s mother earns $150,000 a year. How much do they pay in FICA taxes? Evelyn’s father SS = 6.2%(57,600) = 3571.20 Medicare = 1.45%(68,000) = 986.00 FICA = $4557.20 Hari’s mother SS = 6.2%(57,600) = 3571.20 Medicare = 1.45%(135,000) = 1957.50 FICA = $5528.70
8. 4-2Fluctuating Factors of Production Warm-up Word Definition Symbol/Formula Add-on Costs Commodity Utility
9. 4-2Fluctuating Factors of Production Skill 1: Production costs Example Evelyn, Greg, Freda, and Hari want to find the costs involved in producing and selling hand-painted 12 t-shirts and 12 sweatshirts. They have the following information. Materials $5.50/t-shirts $7.50/sweatshirt paints $1.25/shirt Labor $5/hr Evelyn & Greg, 15 hrs each Freda & Hari, 10 hrs each Packaging small plastic bags, $2/100
10. 4-2Fluctuating Factors of Production Example cont. Evelyn, Greg, Freda, and Hari want to find the costs involved in producing and selling hand-painted 12 t-shirts and 12 sweatshirts. They have the following information. Advertising flyers $5/100 Energy part of utility bill at Evelyn’s house, $2/week Transportation Freda’s car, $25/week What is the total cost to make 12 t-shirts and 12 sweatshirts in on week?
11. 4-2Fluctuating Factors of Production Skill 2: Establishing prices Example Evelyn, Greg, Freda, and Hari want to price the hand-painted 12 t-shirts and 12 sweatshirts so they could cover all their costs. What should be the price of the t-shirts and sweatshirts to pay all costs? The cost to produce 24 shirts from example 1 is $468.48. Therefore each shirt should cost 468.48 ÷ 24 = $19.52 The average of the 2 shirt prices should be $20.
12. 4-3 Cost, Revenue, and Profit Functions Warm-up Word Definition Symbol/Formula Competition Cost Function Fixed Costs Profit (function) Revenue (function) Unit Cost Unit Price Variable Costs
13. 4-3 Cost, Revenue, and Profit Functions Skill 1: Cost function Cost Function c = un + f where c = total cost u = unit cost n = number of units f = fixed cost Example 1 If students purchase and paint n shirts at a cost of $6 each, then the unit cost is $6. What is the cost function assuming there are no fixed costs? c = un + f c = 6n
14. 4-3 Cost, Revenue, and Profit Functions Example 2 Evelyn and her friends have decided to work a fixed number of hours and to start their business by doing only t-shirts. The labor, advertising, energy and transportation are included in the fixed costs which is $282. The variable costs includes materials for production and packaging per t-shirt and it is $6.77. What is the cost function for the business? c = un + f c = 6.77n + 282
15. 4-3 Cost, Revenue, and Profit Functions Skill 2: Revenue function Revenue Function r = snwhere r = revenue s = unit price n = number of units sold Example 1 N shirts are sold at $22 each. What is the revenue function? r = sn r = 22n
16. 4-3 Cost, Revenue, and Profit Functions Example 2 Evelyn and her friends have decided to work a fixed number of hours and to start their business by doing only t-shirts. The labor, advertising, energy and transportation are included in the fixed costs which is $282. The variable costs includes materials for production and packaging per t-shirt and it is $6.77. With a selling price of $19, what is the revenue function? r = sn r = 19n
17. 4-3 Cost, Revenue, and Profit Functions Skill 3: Profit function Profit Function p = r –c where p = profit r = revenue c = total cost Example 1 If students purchase and paint n shirts at a cost of $6 each, then the unit cost is $6. Shirts are sold at $22 each. What is the profit function? p = r - c p = sn – (un + f) p = 22n – 6n = 16n
18. 4-3 Cost, Revenue, and Profit Functions Example 2 Evelyn and her friends have decided to work a fixed number of hours and to start their business by doing only t-shirts. The labor, advertising, energy and transportation are included in the fixed costs which is $282. The variable costs includes materials for production and packaging per t-shirt and it is $6.77. With a selling price of $19, what is the profit for selling 30 t-shirts? p = r - c p = sn – (un + f) p = 19(30) – [6.77(30) + 282] p = 570 – 485.10 p = $84.90
19. 4-3 Cost, Revenue, and Profit Functions Example 3 Using the previous example, Hari wants to compare profits for 15 t-shirts and for 35 t-shirts. For 15 shirts: For 35 shirts: p = r – c p = r – c p = sn – (un + f) p = sn – (un + f) p = 19(15) – [6.77(15) + 282] p = 19(35) – [6.77(35) + 282] p = 285 – 383.55 p = 665 – 518.95 p = - $98.55 p = $146.05 There was a loss of $98.55.
20. 4-4 Break-even Point Warm-up Word Definition Symbol/Formula Break-even Point Loss Region Profit Region
21. 4-4 Break-even Point Skill 1 : Break-even point Break-even point is found by solving a system of equations in which one is the cost function and the other is the revenue function. x = number of items sold y = cost of selling x items
22. 4-4 Break-even Point Example Hari wants to find the break-even point for the t-shirt business when the fixed cost is $282, the unit cost is $6.77, and the selling price is $19. How can Hari find the break-even point algebraically? Cost equation: c = un + f Revenue equation: r = sn y = 6.77x + 282 y = 19x Let 6.77x + 282 = 19x 23.06 = x y = 19(23.06) y = 438.1
23. 4-4 Break-even Point Example cont. Cost if x = 23 if x = 24 y = 6.77x + 282 y = 6.77x + 282 y = 6.77(23) + 282 y = 6.77(24) + 282 y = 437.71 y = 444.48 Revenue if x = 23 if x = 24 y = 19(23) y = 19(24) y = 437 y = 456 Loss if x = 23 if x = 24 437 – 432.71 = -0.71 456 – 444. 48 = 11.52 24 shirts would be the break-even point.
24. 4-5 Linear Programming Warm-up Word Definition Symbol/Formula Constraints Inventory Linear Programming Maximize Minimize Objective Function
25. 4-5 Linear Programming Skill 1 : Constraints Constraints can be written as inequalities from verbal expressions. Test points can be used in solving the inequalities to find the minimum and maximum of certain factors.
26. 4-5 Linear Programming Example 1 Evelyn and her family are planning a car trip that will take several days. How can you use linear programming to write the following constraints as inequalities and show different ways to make the trip? Let x represent hours and y represent miles. On the first day they are leaving at about noon and don’t want to drive after dark. They have only 7 hours in which to drive. The trip must be no more than 7 hours… x ≤ 7 or x ≥ 0 [x must be greater than 0, since time is positive.]
27. 4-5 Linear Programming Example cont. Evelyn and her family are planning a car trip that will take several days. How can you use linear programming to write the following constraints as inequalities and show different ways to make the trip? Let x represent hours and y represent miles. b. They want to travel at least 300 miles that day. The distance must be at least 300 miles… y ≥ 300
28. 4-5 Linear Programming Example 1 cont. Evelyn and her family are planning a car trip that will take several days. How can you use linear programming to write the following constraints as inequalities and show different ways to make the trip? Let x represent hours and y represent miles. c. They want to stay within the speed limit of 55 miles per hour. The speed must be no more than 55mph… Distance = rate(time) y ≤ 55x
29. 4-5 Linear Programming System 1: x = 7, y = 300 (7, 300) means that they could drive 300 miles in 7 hours, with a speed of ≈ 43 mph Example 1 cont. Solve each pair of equations to determine distance and time traveled for the trip.
30. 4-5 Linear Programming System 2: x = 7, y = 55x y = 55(7) = 385 (7,385) means that they could drive 385 miles in 7 hours, with a speed of 55mph Example 1 cont. Solve each pair of equations to determine distance and time traveled for the trip.
31. 4-5 Linear Programming System 3: y = 300 = 55x 5.45 = x (5.45,300) means that they could drive 300 miles in 5hr 27min, with a speed of 55mph Example 1 cont. Solve each pair of equations to determine distance and time traveled for the trip.
32. 4-5 Linear Programming Example 2 Evelyn, Freda, Greg, and Hari have begun selling both t-shirts and sweatshirts. They purchase t-shirts for $5.50 and the sweatshirts for $7.50. They become unsuccessful but have also discovered constraints that affect their business. How can they use linear programming to find the lowest costs within the given constraints? The objective function to minimize cost is: c = 5.50x + 7.50y where c = the cost x = the number of t-shirts y = the number of sweatshirts
33. 4-5 Linear Programming Example 2 cont. Evelyn, Freda, Greg, and Hari have begun selling both t-shirts and sweatshirts. They purchase t-shirts for $5.50 and the sweatshirts for $7.50. They become unsuccessful but have also discovered constraints that affect their business. How can they use linear programming to find the lowest costs within the given constraints? To satisfy the demand and not disappoint customers, they must produce at least 52 shirts a week. They must sell 52 or more shirts… x + y ≥ 52 y ≥ -x + 52
34. 4-5 Linear Programming Example 2 cont. Evelyn, Freda, Greg, and Hari have begun selling both t-shirts and sweatshirts. They purchase t-shirts for $5.50 and the sweatshirts for $7.50. They become unsuccessful but have also discovered constraints that affect their business. How can they use linear programming to find the lowest costs within the given constraints? b. Their supplier can supply them with no more than 45 blank sweatshirts a week. They can obtain no more than 45 sweatshirts… y ≤ 45
35. 4-5 Linear Programming Example 2 cont. Evelyn, Freda, Greg, and Hari have begun selling both t-shirts and sweatshirts. They purchase t-shirts for $5.50 and the sweatshirts for $7.50. They become unsuccessful but have also discovered constraints that affect their business. How can they use linear programming to find the lowest costs within the given constraints? c. Because of the coming cool weather, they must be prepared to sell at least as many sweatshirts as t-shirts, and possibly more. They will sell at least as many sweatshirts as t-shirts… y ≥ x
36. 4-5 Linear Programming Example 2 cont. Evelyn, Freda, Greg, and Hari have begun selling both t-shirts and sweatshirts. They purchase t-shirts for $5.50 and the sweatshirts for $7.50. They become unsuccessful but have also discovered constraints that affect their business. How can they use linear programming to find the lowest costs within the given constraints? d. Because of the time available, they cannot prepare more than 70 shirts of both kinds per week. They cannot prepare more than 70 shirts… y + x ≤ 70 y ≤ -x + 70
37. 4-5 Linear Programming System 1: y = -x + 52, y = 45 -x + 52 = 45 x = 7 (7,45) means that they can make 7 t-shirts and 45 sweatshirts; a total of 52 shirts. C = 5.50(7) + 7.50 (45) = $376 Example 2 cont. Solve each pair of equations to determine the number of t-shirts (x) and sweatshirts (y) they could produce. Then find the cost for producing them.
38. 4-5 Linear Programming System 2 y = -x + 52, y = x -x + 52 = x x = 26 (26,26) means that they can make 26 t-shirts and 26 sweatshirts; a total of 52 shirts. C = 5.50(26) + 7.50(26) = $338 Example 2 cont. Solve each pair of equations to determine the number of t-shirts (x) and sweatshirts (y) they could produce. Then find the cost for producing them.
39. 4-5 Linear Programming System 3 y = x, y = -x + 70 x = -x + 70 x = 35 (35,35) means that they can make 35 t-shirts and 35 sweatshirts; a total of 70 shirts. C = 5.50(35) + 7.50(35) = $455 Example 2 cont. Solve each pair of equations to determine the number of t-shirts (x) and sweatshirts (y) they could produce. Then find the cost for producing them.
40. 4-5 Linear Programming System 4 y = 45, y = -x + 70 45 = -x + 70 x = 25 y = -25 + 70 = 45 (25,45) means that they can make 25 t-shirts and 45 sweatshirts; a total of 70 shirts. C = 5.50(25) + 7.50(45) = $475 The lowest cost within the given constraints would be achieved from making 26 of each shirt. Example 2 cont. Solve each pair of equations to determine the number of t-shirts (x) and sweatshirts (y) they could produce. Then find the cost for producing them.
41. 4-5 Linear Programming Example 3 Evelyn’s group can raise some additional money to invest, so they want to maximize profit. They sell the t-shirts for $17 each and the sweatshirts for $23 each. How can they find the sales quantities that will give the maximum profit within the given constraints? The revenue function is r = 17x + 23y where r = the revenue x = the number of t-shirts y = the number of sweatshirts
42. 4-5 Linear Programming System 1: (7, 45) r = 17x + 23y = 17(7) + 23(45) = $1154 p = r – c = 1154 - 376 = $778 Example 3 cont. Using the same points as in example 2, find the revenue. Then use the cost from example 2 to find the profit.
43. 4-5 Linear Programming System 2: (26,26) r = 17x + 23y = 17(26) + 23(26) = $1040 p = r – c = 1040 - 338 = $702 Example 3 cont. Using the same points as in example 2, find the revenue. Then use the cost from example 2 to find the profit.
44. 4-5 Linear Programming System 3: (35,35) r = 17x + 23y = 17(35) + 23(35) = $1400 p = r – c = 1400 - 455 = $945 Example 3 cont. Using the same points as in example 2, find the revenue. Then use the cost from example 2 to find the profit.
45. 4-5 Linear Programming System 4: (25,45) r = 17x + 23y = 17(25) + 23(45) = $1460 p = r – c = 1460 - 475 = $985 The greatest profit within the given constraints can be made by selling 25 t-shirts and 45 sweatshirts. Example 3 cont. Using the same points as in example 2, find the revenue. Then use the costs from example 2 to find the profit.