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Excalidraw Data
Text Elements
river
f = x + 2y
xy = 125000
y = 125000/x
F = x + 2(125000/x)
F’ = 1 - 250,000x^-2 0 = 1 - 250000/x^2
x^2 = 250000 x = 500
1^2-250000 = - 1000^2 - 250000 = + 500 is a min
125000/500 = 250
area = 989 xy = 989 y = 989/x Cost = 20.70(2x) + 20.70(2y) + 18(2x) C = 41.4x + 41.4y + 36x C = 77.4x + 41.4y C = 77.4x + 41.4(989/x) C = 77.4x + 40944.6/x C’ = 77.4x + 40944.6x^-2 0 = 77.4x + 40944.6x^-2 0 = (77.4^2 + 40944.6)/x^2 0 = 77.4x^2 - 40944.6 40944.6 = 77.4x^2 529 = x^2 23 = x
23 is a min 989/23 = 43 23,43
x
x
y
80 = x^2 * y cost = 2(.60x^2) + 4(.30xy) cost = 1.2x^2 + 1.2xy y = 80/x^2 cost = 1.2x^2 + 1.2x * (80/x^2) cost = 1.2x^2 + 96/x cost’ = 2.4x - 96x^-2 -2.4x = -96x^-2 -2.4 = -96x^-3 1 = 40x^-3 x^3 = 40 x = cbrt(40) x = 3.42
3.42 is a min 80/(3.42)^2 6.83
3.42, 6.84
40
15 miles
Cost = 328000(sqrt(x^2+40^2)) + 72000(15-x) C = 328000(x^2+40^2)^(1/2) + 1080000 - 72000x C’ = 328000(1/2(x^2+40^2)^(-1/2)(2x)) - 72000 C’ = (328000x)/(x^2+40^2)^(1/2) - 72000 C’ = 328000x/(sqrt(x^2+40x^2)) - 72000 0 = (328000x)/(x^2+40x^2)- (72000*x^2+40^2)/sqrt(x^2+40^2) 0 = 328000x - 72000sqrt(x^2+40^2)/sqrt(x^2+40^2) 0 = 328000x - 72000sqrt(x^2+40^2) 72000sqrt(x^2+40^2) = 328000x sqrt(x^2+40^2) = (41/9)x x^2 + 40^2 = (1681/81)x^2 40^2 = (1600/81)x^2 81 = x^2 9 = x
8t^2 + 40t
16t + 40
16(6) + 40 = 136
bl
h
sA = bl^2 + 4(h*bl) Volume = bl^2 * h 4000 = bl^2 * h 4000/bl^2 = h sA = bl^2 + 4(4000/bl^2 * bl) sA = bl^2 + 16000/bl sA = bl^2 + 16000bl^-1 sA’ = 2bl - 16000bl^-2 0 = 2bl - 16000bl^-2 16000bl^-2 = 2bl 16000/bl^2 = 2bl 16000 = 2bl^3 8000 = bl^3 20 = bl
sA = 20^2 + 16000/20 sA = 20^2 + 800 sA = 400 + 800 sA = 1200
Volume = x^2 * h 4840 = x^2 * h 4840/x^2 = h 4840/h = x
11(4(x(4840/x^2))) + 9x^2 + x^2 44(x(4840/x^2)) + 9x^2 + x^2 44x(4840/x^2) + 10x^2 44x(4840x^-2) + 10x^2 212960x^-1 + 10x
10x^2 + 212960x^-1 20x - 212960x^-2 0 = 20x - 212960x^-2 212960/x^2 = 20x 212960 = 20x^3 10648 = x^3 22 = x
h = 4840/x^2 h = 4840/22^2 h = 4840 / 484 h = 10
11/ft
9/ft
1/ft
v = 15 b = x * 3x 15 = b * h 15 = x * 3x * h 15 = 3x^2 * h 15/3x^2 = h h = 15/3x^2 h = 5/x^2 b = 3x
sidepair1 = 3(5/x) sidepair1 = 15/x sidepair2 = x(5/x^2) sidepair2 = 5/x sides = 2(15/x) + 2(5/x) baseandtop = 2(3x^2) baseandtop = 6x
cost = 2.2(6x^2) + 2.4(2(15/x) + 2(5/x)) cost = 13.2x^2 + 4.8(15/x) + 4.8(5/x) cost = 13.2x^2 + 72/x + 24/x cost = 13.2x^2 + 96/x cost’ = 26.4x - 96x^-2 cost’ = 26.4x - 96/x^2 0 = 26.4x - 96/x^2 96/x^2 = 26.4x 96 = 26.4x^3 96/26.4 = x^3 cbrt(96/26.4) = x cbrt(40/11) = x
h = 5/x^2 h = 5/(cbrt(40/11)^2) h = 2.11
C = 2.2(2(3x^2)) + 2.4(2(3x^2*(5/x^2))) C = 2.2(6x^2) + 9.6(3x^2(5/x^2)) C = 13.2x^2 + 28.8x^2(5/x^2) C = 13.2x^2 + 28.8x^25x^-2 C = 13.2x^2 + 144x^2 * x^-2 C = 13.2x^2 + 144 C’ = 26.4x 0 = 26.4x 0 = x
3956 = x * y y = 3956/x
outside = 2x+2y outside = 2x + 2(3956/x) inside = 2x
cost = 11.5(outside) + 10(inside) cost = 11.5(2x+2(3956/x) + 10(2x) cost = 11.5(2x+(7912/x)) + 20x cost = 23x + 90988/x + 20x cost = 43x + 90988/x cost’ = 43 + 90988x^-2 0 = 43 + 90988x^-2 90988x^-2 = 43 90988 = 43x^2 2116 = x^2 46 = x
3956/46 86 = y
90
86
188ft
x + y = 188 y = 188-x
area = x * (188-x) area = 188x - x^2 area’ = 188 - 2x 0 = 188 - 2x 2x = 188 x = 188/2 x = 94 188 - 94 = 94
x = 94 y = 94
cant exceed 86 and 90 though
must extend the walls!
x
x-90
90
86
y-86
y
area = (86 + y-86)(90 + x-90) 188 = (y-86) + (x-90) + x + y 188 = 2x + 2y - 176 364 = 2x + 2y 182 = x + y y = 182-x area = x * (182-x) area = 182x - x^2 area’ = 182 - 2x 2x = 182 x = 91
182 - 91 = 91 91*91 = 8281
-11t^2 + 352t + 7755 -22t + 352
0 = -22t + 352 22t = 352 t = 16
-11(16)^2 + 352(16) + 7755 -11(256) + 5632 + 7755 -11(256) + 13387 -2816 + 13387 10571
-11t^2 + 352t + 7755 t^2 - 32t - 705 (x+15)(x-47) zeros at -15 and +47 projectile hit the ground at 47.
-705
but add to -32
15 and -47
solve derivative for 47
-22t + 352 -22(47) + 352 -1034 + 352 -682
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