os ando nd how to obtato the problem An t tn foAnwrg h Ill. Tangent Lines and Op

ID: 1155500 • Letter: O

Question

os ando nd how to obtato the problem An t tn foAnwrg h Ill. Tangent Lines and Optimization In business and economics, common questions of interest include or revenue, how to minimize average costs, and how to maxi For questions such as these, the description of how to obtai minimum represents an optimal (or best possible) solution to t desiredPr cess of finding an optimal solution is called optimization. Ans roblen often involves tangent lines. In this project, we examine howt how to maximiue mony obtain the problem. And the ph aver of how to obtain the tangent lines Suppose that Wittage, Inc., manufactures paper shredders and that its weekly total costs (in dollars) for x shredders are given thiome ad for home and effice given minimize average costs Clx) 0.03x +12.75x+6075 The average cost per unit for x units Idenoted by Cl)] is the total cost diy ber of units. Thus, the average cost function for Wittage, Inc., is Cla) 0.03x +12.75x + 6075 0.03x 12.75x 6075 C(x)0.03x + 12.75 Wittage, Inc, would like to know how the company can use marginal costs to mation about minimizing average costs. To investigate this relationshigis Inie following questions. ate this relationship, answer the 1. Find Wittage's marginal cost function, then complete the following table. 100 200 300 400 500 600 C (x) MC (x)

Explanation / Answer

Given Total cost function= C(x) = 0.03x2 + 12.75x + 6075

Total cost is the sum of variable cost and fixed cost. Average cost equals to the total cost divided by the no of goods produced. So in another words average cost is the sum of average fixed cost and average variable cost.

So, average cost = TC/ quantity = (0.03x2 + 12.75x + 6075)/ x = 0.03x + 12.75 + 6075/x

So, for 100 unit, AC = 0.03x + 12.75 + 6075/x = 0.03*100 + 12.75 + 6075/100 = 76.5

For 200 unit, AC = 0.03x + 12.75 + 6075/x = 0.03*200 + 12.75 + 6075/200 = 49.13

For 300 unit, AC = 0.03x + 12.75 + 6075/x = 0.03*300 + 12.75 + 6075/300 = 42

For 400 unit, AC = 0.03x + 12.75 + 6075/x = 0.03*400 + 12.75 + 6075/400 = 39.94

For 500 unit, AC = 0.03x + 12.75 + 6075/x = 0.03*500 + 12.75 + 6075/500 = 39.90

For 600 unit, AC = 0.03x + 12.75 + 6075/x = 0.03*600 + 12.75 + 6075/600 = 40.88

Marginal cost is the cost of producing one extra unit of good. It include all the cost that is change with the change in production. So, in another words marginal cost is not related with fixed cost , it is related with variable cost.

Given Total cost function= C(x) = 0.03x2 + 12.75x + 6075

In this equation 6075 is the fixed cost that is not changing with level of production. So, considering only variable cost.

Marginal cost = Change in cost for producing one extra unit of good = dTC(q)/dq= 0.06x + 12.75

So, for 100 unit, MC = 0.06x + 12.75 = 0.06*100 + 12.75 = 18.75

So, for 200 unit, MC = 0.06x + 12.75 = 0.06*200 + 12.75 = 24.75

So, for 300 unit, MC = 0.06x + 12.75 = 0.06*300 + 12.75 = 30.75

So, for 400 unit, MC = 0.06x + 12.75 = 0.06*400 + 12.75 = 36.75

So, for 500 unit, MC = 0.06x + 12.75 = 0.06*500 + 12.75 = 42.75

So, for 600 unit, MC = 0.06x + 12.75 = 0.06*600 + 12.75 = 48.75

So, the table will be as below:

100

200

300

400

500

600

Average Cost, C(x)

76.50

49.13

42.00

39.94

39.90

40.88

Marginal Cost MC(x)

18.75

24.75

30.75

36.75

42.75

48.75

When firm’s MC is increasing it means production of every extra unit is costing more than the previous unit. Initially the MC is lower than AC because of fixed cost. But gradually it is increasing. When MC = AC firm is producing at optimal cost. So, when MC is greater than AC firm will decide not to produce. At quantity 500 firm’s MC is greater than AC. So, firm should not produce after 400 units.

100

200

300

400

500

600