SOLUTION: A product may be made using Machine I or Machine II. The manufacturer estimates that the monthly fixed costs of using Machine I are $19,000, whereas the monthly fixed costs of usin

Question 1141850: A product may be made using Machine I or Machine II. The manufacturer estimates that the monthly fixed costs of using Machine I are $19,000, whereas the monthly fixed costs of using Machine II are $15,000. The variable costs of manufacturing 1 unit of the product using Machine I and Machine II are $13 and $19, respectively. The product sells for $60 each.
(a) Find the cost functions C(x) associated with using each machine, where x is the number of units produced using that machine.
Machine I C1(x) =

Machine II C2(x) =

(b) Sketch the graphs of the cost functions of part (a) and the revenue function on the same set of axes. (Sketch the graphs using rays.)
(c) Which machine should management choose to maximize their profit if the projected sales are 450 units?
Machine I
Machine II
Which machine should management choose to maximize their profit if the projected sales are 550 units?
Machine I
Machine II
Which machine should management choose to maximize their profit if the projected sales are 650 units?
Machine I
Machine II
(d) What is the profit for each case in part (c)?
450 units $
550 units $
650 units $
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
cost function for machine 1 is 19000 + 13 * x.

cost function for machine 2 is 15000 + 19 * x.

revenue function is 60 * x

x represents the number of units produced and sold.

when 450 units are sold ........,

revenue is 60 * 450 = 27,000
cost using machine 1 is 19000 + 13 * 450 = 24,850
cost using machine 2 is 15000 + 19 * 450 = 23,550

profit using machine 1 is 27,000 - 24,850 = 2,150
profit using machine 2 is 27,000 - 23,550 = 3,450

profit using machine 2 is more than profit using machine 1.

when 550 units are sold .........,

revenue is 60 * 550 = 33,000
cost using machine 1 is 19000 + 13 * 550 = 26,150
cost using machine 2 is 15000 + 19 * 550 = 25,450

profit using machine 1 is 33,000 - 26,150 = 6,850
profit using machine 2 is 33,000 - 25,450 = 7,550

profit using machine 2 is more than profit using machine 1.

when 650 units are sold .............,

revenue is 60 * 650 = 39,000
cost using machine 1 is 19000 + 13 * 650 = 27,450
cost using machine 2 is 15000 + 19 * 650 = 27,350

profit using machine 1 is 39,000 - 27,450 = 11,550
profit using machine 2 is 39,000 - 27,350 = 11,650

profit using machine 2 is more than profit using machine 1.

profit is equal to revenue minus cost.
if the revenue is the same regardless of which machine is used, then the greater profit is given to the machine that uses the least cost.

the break even point where the cost for using machine 1 versus using machine 2 is found by making their cost functions equal to each other.

you get 19000 + 13 * x = 15000 + 19 * x
subtract 15000 from both sides of this equation and subtract 13 * x from both sides of this equation to get 19000 - 15000 = 19 * x - 13 * x.
combine like terms to get 4000 = 6 * x
solve for x to get x = 666.67 units.
with that number of units, the cost of using machine 1 is the same as the cost of using machine 2.
when the number of units produced and sold are greater than that, machine 1 provides more profit.

for example, when 750 units are produced .........,
revenue is 60 * 750 = 45,000
cost for machine 1 is 19000 + 13 * 750 = 28,750
cost for machine 2 is 15000 + 19 * 750 = 29,250
profit for machine 1 is 45,000 - 28,750 = 16,250
profit for machine 2 is 45,000 - 29,250 = 15,750

profit using machine 1 is more than profit using machine 2.

the following graph shows revenue and cost at 450, 550, 650, and 750 units.

graph of revenue is green.
graph of cost for machine 1 is red.
graph of cost for machine 2 is blue

$$$

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