A. Break-Even Analysis. A multimedia company produces DVDs. One-time fixed costs for a particular DVD are $48,000, which include costs such as filming, editing, and promotion. Variable costs amount to $12.40 per DVD and include manufacturing. packaging, and distribution costs for each DVD actually sold to a retailer. The DVD is sold to retail outlets at $17.40 each. How many DVDs must be manufactured and sold in order for the company to break even? Step 1. Let x= number of DVDs manufactured and sold. Step 2. C= cost of producing x DVDs R= revenue (return) on sales of x DVDs Fixed costs =$48,000 Variable costs =$12.40x C= Fixed costs + variable costs =$48,000+$12.40x R=$17.40x Step 3. The company breaks even if R=C; that is, if $17.40x=$48,000+$12.40x Step 4. 17.4x=48,000+12.4x 5x=48,000x=9,600 Subtract 12.4x from both sides. Divide both sides by 5 . The company must make and sell 9,600 DVDs to break even. 1.a How many DVDs would a multimedia company have to make and sell to break even if the fixed costs are $36,000, variable costs are $10.40 per DVD, and the DVDs are sold to retailers for $15.20 each? B. Cost Equation The management of a company that manufactures skateboards has fixed costs (costs at 0 output) of $300 per day and total costs of $4,300 per day at an output of 100 skateboards per day.Assume that cost C is linearly related to output x. (A) Find the slope of the line joining the points associated with outputs of 0 and 100 ; that is, the line passing through (0,300) and (100,4,300). (B) Find an equation of the line relating output to cost. Write the final answer in the form C=mx+b Solution (A) : m=x2x1y2y1=10004,300300=1004,000=40 (B) We must find an equation of the line that passes through (0,300) with slope 40 . We use the slope-intercept form: c=mx+bC=40x+300 2.a Answer parts (A) and (B) in Example 7 for fixed costs of $250 per day and total costs of $3,450 per day at an output of 80 skateboards per day. C. Supply and Demand At a price of $9.00 per box of oranges, the supply is 320,000 boxes and the demand is 200,000 boxes. At a price of $8.50 per box, the supply is 270,000 boxes and the demand is 300,000 boxes. (A) Find a price supply equation of the form p=mx+b, where p is the price in dollars and x is the corresponding supply in thousands of boxes. (B) Find a price demand equation of the form p=mx+b, where p is the price in dollars and x is the corresponding demand in thousands of boxes. (C) Graph the price supply and price demand equations in the same coordinate system and find their point of intersection. (A) To find a price supply equation of the form p=mx+b, we must find two points of the form (x,p) that are on the supply line. From the given supply data, (320,9) and (270, 8.5) are two such points. First, find the slope of the line: m=x2x1p2p1=32027098.5=500.5=0.01 Now use the point-slope form to find the equation of the line: pp1=m(xx1)p9=0.01(x320)(x1,p1)(320,9)p9=0.01x3.2p=0.01x+5.8 Price-supply equation (B) From the given data, (200,9) and 8.5) are two points on the demand line: demand (300, m=x2x1p2p1=3002008.59=1000.5=0.005pp1=m(xx1)p9=0.005(x200).(x1,p1)(200,9)p9=0.005x+1p=0.005x+10Pricedemandequation (C) From part (A), we plot the points (320,9) and (270,8.5) and then draw the line through them. We do the same with the points (200,9) and (300,8.5) from part (B) (Fig. 7). (Note that we restricted the axes to intervals that contain these data points.) To find the intersection point of the two lines, we equate the right-hand sides of the price supply and price demand equations and solve for x : Price-supply Price-demand 0.01x+5.80.015xx=0.005x+10=4.2=280 Now use the price supply equation to find p when x=280 : p=0.01x+5.8p=0.01(280)+5.8=8.6 The lines intersect at (280,8.6). The intersection point of the price supply and price demand equations is called the equilibrium point, and its coordinates are the equilibrium quantity (280) and the equilibrium price ($8.60). These terms are illustrated in Figure 7. 3. A At a price of $12.59 per box of grapefruit, the supply is 595,000 boxes and the demand is 650,000 boxes. At a price of $13.19 per box, the supply is 695,000 boxes and the demand is 590,000 boxes. Assume that the relationship between price and supply is linear and that the relationship between price and demand is linear. (A) Find a price supply equation of the form p=mx+b. (B) Find a price demand equation of the form p=mx+b (C) Find the equilibrium point