a rectangular sheet of metal measures 50 cm by 40 cm.Each squares of side x cm are cut from each corner and discarded.The sheet is then folded up to make a tray of depth x cm. What is the domain of possible values of x? Find the possible values of x which maximizes the capacity of the tray.
2007-10-08 22:42:51 · 3 answers · asked by fm 1 in Science & Mathematics ➔ Mathematics
maximizing the capacity means maximizing the volume
V=x(50-2x)(40-2x)
V=x(2000-80x-100x+4x^2)
V=x(2000-180x+4x^2)
V=2000x-180x^2+4x^3
differentiate V in terms of x
dV/dx=2000-180+12x^2
maximize dV/dx
dV/dx=0
2000-180+12x^2=0
1000-90x+6x^2=0
x1=7.5 cm
x2=7.5 cm
2007-10-08 22:53:39 · answer #1 · answered by ptolemy862000 4 · 0⤊ 0⤋
V (x) = (50 - 2x)(40 - 2x) (x)
V(x) = 4 x ³ - 180 x ² + 2000 x
V `(x) = 12 x ² - 360 x + 2000
3 x ² - 90 x + 500 = 0 for max V
x = [ 90 ±√(8100 - 6000)] / 6
x = [ 90 ±√(2100) ] / 6
x = [ 90 ± 45.8) ] / 6
x = 22.6 cm , x = 7.37 cm are possible values.
2007-10-08 23:47:25 · answer #2 · answered by Como 7 · 0⤊ 0⤋
x shall be 7.36238cm
2007-10-08 23:06:34 · answer #3 · answered by skraj 1 · 0⤊ 0⤋