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find production level that minimizes average CPM production level means find x= average cost = c/x
ac = (800+.04x+.0002x^2 ) / x ac = 800/x + .04 + .0002 ac = -800x^-2 +.0002 0 = -800/x^2 +.0002 0 = (-800+.0002x^2) / x
-800 + .000x^2 = 0 .0002x^2 = 800 x^2 = 4000000 x = 2000 neg pos 2000 is a minimum
demand = 14 cost = .2x^2 + 6.4x + 2 profit = revenue - cost p = 14x- (.2x^2 + 6.4x + 2) p = 14x - .2x^2 -6.4x - 2 p = -.2x^2 + 7.6x - 2 p’ = -.4x + 7.6 0 = -.4x + 7.6 -7.6 = -.4x 19 = x pos neg, 19 is a max.
p(19) = -.2(19)^2 + 7.6(19) -2 70.2
THEY CHANGED THE NUMBERRS SINCE THIS PDF CAME OUT
really solving 0.2x^2 + 6.4x + 2
C(x) = 6 * (x/2) + 60 * (500/x) C(x) = 6x/2 + 30000/x C(X) = 3x + 30000/x C(X) = 3x + 30000x^-1 C’(X) = 3 - 30000x^-2 3-30000x^-2 = 0 3-(30000/x^2) = 0 -(30000/x^2) = -3 -30000 = -3x^2 10000 = x^2 100 = x
neg pos min at 100
lot size 100 500/100 = 5 number of orders 5
120 bracelets total 2.40 storage cost fixed cost per order 16 2.60 per item ordered
C(x) = 2.40 * x/2 + (16 + 2.60x) * (120/x) C(x) = 1.20x + (16 + 2.60x) * ( 120x^-1 ) C(x) = 1.20x + 1920x^-1 + 312 C’(X) = 1.20 - 1920x^-2 0 = 1.20 - 1920x^-2 -1.20 = -1920/x^2 -1.20x^2 = -1920 x^2 = 1600 x = 40 neg pos 40 is a min 120/40 = 3 3 orders of 40.
100 = 90x
demand = 100-5x
R = price * how many we sell
how many we sell = 100-5(price)
R = (90+5x) * (100-5x)
R = 9000 + 500x -450x -25x^2
R = 9000 - 25x^2 + 50x
R’ = -50x + 50
0 = -50x + 50
-50 = -50x
1 = x
R” = -50
concave down
x = 1 is a max
95$ per tennis racket maximizes revenue
120textbooks / yr 4.80 to store for a year 8$ per order
C(X) = 4.8 * x/2 + 8 * ( 120 / x ) C(X) = (4.8x)/2 + 960/x C(x) = 2.4x + 960x^-1 C’(x) = 2.4 - 960x^-2 C’(x) = 2.4 - 960/x^2 0 = 2.4 - 960/x^2 960/x^2 = 2.4 960 = 2.4x^2 400 = x^2 20 = x
120/20 = 6
C(X) = 1.40 * x/2 + 17.5 * ( 400 / x ) C(x) = .7x + 7000/x C(x) = .7x + 7000x^-1 C’(X) = .7 - 7000x^-2 C’(x) = .7 - 7000/x^2 0 = .7 - 7000/x^2 7000/x^2 = .7 7000 = .7x^2 10000 = x^2 100 = x
400/100 = 4
C(x) = 14 * x/2 + 63 * ( 81/x ) C(x) = 7x + 5103/x C(x) = 7x + 5103x^-1 C’(x) = 7 - 5103x^-2 C’(x) = 7 - 5103/x^2 0 = 7 - 5103/x^2 5103/x^2 = 7 5103 = 7x^2 729 = x^2 27 = x
81/27 = 3
x is how many 6 dollar increases in price we do price per book = 60 + 6(x) quantity sold of book per month = 240 - 6(x) revenue = price * quantity revenue = (60 + 6(x)) * (240-6(x)) revenue = 14400 + 1440x - 360x -36x^2 revenue = -36x^2 + 1080x + 14400 revenue’ = -72x + 1080 0 = -72x + 1080 72x = 1080 x = 15
price = 60 + 6(15) 60 + 90 = 150
C(x) = 576 * x/2 + (324 + 257x) * (1152/x) C(x) = 288x + 373248/x + 296064 C(x) = 288x + 373248x^-1 + 296064 C’(x) = 288 - 373248x^-2 0 = 288 - 373248/x^2 373248/x^2 = 288 373248 = 288x^2 1296 = x^2 36 = x
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