Math 1313 Section 1.5 Linear Cost, Revenue And Profit .

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Math 1313Section 1.5Linear Cost, Revenue and Profit Functions:If x is the number of units of a product manufactured or sold at a firm then,The cost function, C(x), is the total cost of manufacturing x units of the product.Fixed costs are the costs that remain regardless of the company’s activity.Examples: building fees (rent or mortgage), executive salariesVariable costs are costs that vary with the production or sales.Examples; wages of production staff, raw materialsThe revenue function, R(x), is the total revenue realized from the sale of x units of the product.The profit function, P(x), is the total profit realized from the manufacturing and sale of the x units ofproduct.Formulas: Suppose a firm has fixed cost of F dollars, production cost of c dollars per unit and sellingprice of s dollars per unit thenC(x) R(x) P(x) Where x is the number of units of the commodity produced and sold.Example 3: A manufacturer has a monthly fixed cost of 150,000 and a production cost of 18 for eachunit produced. The product sells for 24 per unit.a. What is the cost function?b. What is the revenue function?c. What is the profit function?d. Compute the profit (loss) corresponding to production levels of 22,000 and 28,000.e. How many units must the company produce and sell if they wish to make a profit of 40,000?2

Math 1313Section 1.5Popper 2Question 1An office building worth 1 million when completed in 2000 is being depreciated linearly over 50years. (Assume scrap value is 100,000) What is the linear depreciation?a.b.c.d. 20000 18000 2000None of the aboveExample 4: Auto Time, a manufacturer of 24-hour variable timers, has a fixed monthly cost of 56000and a production cost of 10 per unit manufactured. The timers sell for 17 each.a. What is the cost function?b. What is the revenue function?c. What is the profit function?d. Compute the profit (loss) corresponding to the production and sale of 4,000, 8,000 and 10,000 timers.Break-Even PointThe break-even level of operation- is when the company neither makes a profit nor sustains a loss.Note: The break-even level of operation is represented by the point of intersection of two lines.The break-even level of production means the profit is zero.3

Math 1313Section 1.5Consider the following graph.The point (xo , yo ) is referred to as the break-even point.xo break even quantityyo break even revenueIf x xo then R(x) C(x), therefore P(x) R(x) –C(x) 0 so you will have a loss.If x xo then R(x) C(x), therefore P(x) R(x) – C(x) 0 so you will have a profit.Example 5: Find the break-even quantity and break-even revenue if C(x) 110x 40,000 andR(x) 150x.Example 6: The XYZ Company has a fixed cost of 20,000, a production cost of 12 for each unitproduced and a selling price of 20 for each unit produced.a. Find the break-even point for the firm.b. If the company produces and sells 2000 units, would they obtain a profit or loss?4

Math 1313Section 1.5c. If the company produces and sells 3000 units, would they obtain a profit or loss?Popper 2Question 3A division of Carter Enterprises produces ‘‘Personal Income Tax’’ diaries. Each diary sells for 18.The monthly fixed costs incurred by the division are 35,000, and the variable cost of producingeach diary is 4. What is the profit function?a. 22 3500b. 14 35000c. 14 35000d. None of the abovePopper 2Question 4Find the break-even point for the previous problem.a. 8333.33, 25000b. 3125, 25000c. 2500, 45000d. None of the above5

Math 1313Section 1.5Example 7: Given the following profit function P(x) 6x -12,000.a. How many units should be produced in order to realize a profit of 9,000?b. What is the profit or loss if 1,000 units are produced?Example 8: A bicycle manufacturer experiences fixed monthly costs of 124,992 and variable costs of 52 per standard model bicycle produced. The bicycles sell for 100 each. How many bicycles must heproduce and sell each month to break even? What is his total revenue at the point where he breaks even?6

Math 1313Section 2.1Section 2.1: Solving Linear Programming ProblemsDefinitions:An objective function is subject to a system of constraints to be optimized (maximized orminimized)Constraints are a system of equalities or inequalities to which an objective function is subject to.A linear programming problem consists of an objective function subject to a system ofconstraintsExample of what they look like:An objective function is max P(x,y) 3x 2y or min C(x,y) 4x 8y5x 3y 120Constraints are:2x 6 y 60A linear programming problem consists of a both the objective function subject to restraints.Max P ( x , y ) 3x 2 yx y 4St:2 x 5 y 80x, y 0Consider the following figure which is associated with a system of linear inequalities:Definitions:The region is called a feasible set. Each point in the region is a candidate for the solution of theproblem and is called a feasible solution.The point(s) in region that optimizes (maximizes or minimizes) the objective function is called theoptimal solution.Fundamental Theorem of Linear Programming Given that an optimal solution to a linear programming problem exists, it must occur at avertex of the feasible set.If the optimal solution occurs at two adjacent vertices of the feasible set, then the linearprogramming problem has infinitely many solutions. Any point on the line segment joiningthe two vertices is also a solution.1

Math 1313Section 2.1This theorem is referring to a solution set like the one that follows:MaximumMinimumThe Method of Corners1. Graph the feasible set.2. Find the coordinates of all corner points (vertices) of the feasible set.3. Evaluate the objective function at each corner points.4. Find the vertex that renders the objective function a maximum (minimum). If there is only onesuch vertex, then this vertex constitutes a unique solution to the problem. If the objective functionis maximized (minimized) at two adjacent corner points of S, there are infinitely many optimalsolutions given by the points on the line segment determined by these two vertices.Example 1: Given the following Linear Program, Determine the vertices of the feasible set.Max profit P (x , y ) 12x 10y15x 10 y 1380Subject to: 10x 12 y 1320x, y 02

Math 1313Section 2.1Example 2: Given the following Linear Program, Determine the vertices of the feasible setMin D3 3x y10x 2 y 84Subject to: 8x 4 y 120x, y 03

Math 1313 Section 1.5 2 Linear Cost, Revenue and Profit Functions: If x is the number of units of a product manufactured or sold at a firm then, The cost function , C(x), is the total cost of manufacturing x units of the product. Fixed costs are th

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