Application of Mathematics in Business : F 107 - Group K

Application of Mathematics in Business : F 107 - Group K

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  F-107: Business MathematicsII Application of Linear Equations and Break-even Analysis Application of Differential Calculus
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  Group No. ‘K’Section “A”, BBA 15 th Batch Department of Finance University of Dhaka SL Name ID 1 Muhammad Jafar Sadik Chowdhury 15-167 2 Esrat Karim 15-173 3 Archis Banerjee 15-179
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  OBJECTIVES OF THESTUDY BestWay CNG Filling Station By using Linear Equations To determine cost, profit, revenue functions. To determine net profit for BestWay CNG Filling Station before taxes. To analyze Break-Even-Points & Equilibrium Points. By using Differential Calculus To determine the new area of the factory building for BestWay CNG Filling Station . To test the local minima and to determine new value of length and breadth.
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  LINEAR EQUATIONS ANDFUNCTIONS Linear programming is used to solve problems in many aspects of business administration including: product mix planning distribution networks truck routing staff scheduling financial portfolios corporate restructuring A common form : Thus, 2x – 3y = 7 is a linear equation because it consists of the constant 7, the term 2x (which is the constant 2 times x to the first power), and -3y, which is also a term consisting of a constant times one variable to the first power.
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  Slope == = 3 Vertical Change 6 Horizontal Change 2 6 = Vertical change (rise) 2 = Horizontal change (run) B A ● F(1, -1) ● E(1, -3) ● D(7.5, -3) ● K(-5, 6) ● L(-4, 4) ● C(2.5, -7) ● Q(-6, 2) ● J(-2, 2) ● G(-4, -6) ● H(-3, -3) 0 Slope = m = = Difference of y’s y 2 ̶ y 1 Difference of x’s x 2 ̶ x 1
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  Total Revenue = Total Cost  R (Q) = C (Q)  Price x Quantity = Total Fixed Cost + (Variable Cost x Quantity)  P x Q = TFC + (V x Q)  P x Q – (V X Q) = TFC  Q (P – V) = TFC  Break-even Quantity = TFC / (P – V) Break-even Analysis
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  LINEAR EQUATIONS TOSOLVE BUSINESS PROBLEMS Bestway CNG Filling Station expects fixed cost of $22,800. Margin is to be 55% of retail. Variable cost in addition to costs of goods is estimated at $0.17 per dollar of sales. Requirements:
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  So, Cost function= C(x) = 22,800 + 0.17x + 0.45x C(x) = 22,800 + 0.62x Revenue function = R(x) = x So, the profit function = R(x) – C(x) = x – 22,800 – 0.62x P(x) = 0.38x – 22,800 Fixed cost for goods = $ 22,800 Variable cost = 0.17x Margin = 55% of retail Cost of goods sold = Retail Price – Margin = s – 0.55x = 0.45x
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  Break-even Point For Bestway Cng Filling Station According to the theory of Break-Even we know that at the point of break-even, P (x) = 0 Profit function = P(x) = 0  0.38x – 22,800 = 0  x = 60,000  Break-even Quantity = 60,000 So, the break-even dollar volume of sales is, R(60,000) = $ 60,000 So, the break-even point is (60000, 60000) Profit For Bestway Cng Filling Station The net profit before taxes be on sales of 75,000 is, P (75,000) = 0.38 × 75,000 – 22,800 = $5,700
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  Break-even chart analysisfor BestWay CNG Filling Station Revenue and cost 100,000 75,000 50,000 0 50,000 100,000 Sales Break-even point (60,000, 60,000) Cost: C(x) = 0.62x + 22,800 Revenue: R(x) = x
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  DIFFERENTIAL CALCULUS Differentialcalculus, a field in mathematics, is the study of how functions change when their inputs change. The process of finding a derivative is called differentiation. The fundamental theorem of calculus states that differentiation is the reverse process to integration. The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line equals the derivative of the function at the marked point.
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  THE DERIVATIVE Derivativeis how the function changes when its inputs change. where the symbol Δ (the uppercase form of the Greek letter Delta) is an abbreviation for "change in". It follows that Δy = m Δx Change in y  y Change in x  x m = = Tangent line and slope
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  DIFFERENTIAL CALCULUS TOSOLVE BUSINESS PROBLEMS For the Filling Station a floor area of 40,000 square feet is to be built where we require parking place 60 feet wide front and canteen 60 feet wide rear side of the area, and 40 feet wide at either side for warehouse . Find the dimensions and minimum area for Bestway CNG Filling Station that meets these specifications. Since the front and rear side are to be 60 feet each, So the new length is = x + 60 + 60 = x + 120 Since each side is to be 40 feet, so the new breadth is = y + 40 + 40 = y + 80 By assuming first derivative = 0, we find x = 245.4 feet y = 163 feet
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  TESTING WHETHER LOCALMINIMUM Testing y = 163 in , we have = > 0 which is positive. So it is local minimum .
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  The total dimensionwill then be 245.4 + 120 = 365.4 feet by 163 + 80 = 243 feet So, the total area that meets the specifications is: 365.4 × 243 = 88,792.2 square feet So, the new area with parking, canteen and warehouse is to be 88,792.2 sq feet . BestWay CNG Filling Station Canteen y 60 y 60 Parking x 40 x 40 Warehouse Warehouse Parking Canteen
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  FINDINGS AND DECISIONSUsing linear Equations in determining cost, revenue and profit functions. Analyzing of Break-Even-Points & Equilibrium Points by using cost functions. Net profit before taxes is determined by profit function . Differential Calculus is applied in one way to determine the area with warehouse, canteen and parking place . First two derivatives are helpful to determine the value of new length and new breadth . Whether the local minima is tested by second derivative.
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