一份十字绣要裱框,镜框左右两边宽度均为0.225m,上下两边宽度均为0.1m,裱装完的总面积最小达到5.29m2.问十字绣长和宽各是多少米时,十字绣的面积最小

Question

一份十字绣要裱框,镜框左右两边宽度均为0.225m,上下两边宽度均为0.1m,裱装完的总面积最小达到5.29m2.问十字绣长和宽各是多少米时,十字绣的面积最小

Answer 1

(x-0.45)(y-0.2)=5.29,x>0.45,y>0.2
求xy的最小值.
xy-0.2x-0.45y+0.09=5.29
xy=0.2x+0.45y+5.20
设xy=a,y=a/x
a=0.2x+0.45a/x+5.20
a(1-0.45/x)=0.2x+5.20
a=(0.2x+5.20)/(1-0.45/x)=(0.2x^2+5.20x)/(x-0.45)
a'=(0.4x+5.20)-(0.2x^2+5.20x)/(x-0.45)^2
=[(0.4x+5.20)(x-0.45)-(0.2x^2+5.20x)]/(x-0.45)^2
=[0.4x^2+5.02x-2.34-0.2x^2-5.20x]/(x-0.45)^2
=[0.2x^2-0.18x-2.34]/(x-0.45)^2=0
0.2x^2-0.18x-2.34=0
x^2-0.9x-11.7=0
x=[0.9±√(0.9^2+4x11.7)]/2=[0.9±√47.61]/2=[0.9±6.9]/2,x>0,x=(0.9+6.9)/2=3.9
a=(0.2x3.9^2+5.20x3.9)/(3.9-0.45)=6.76 这就是最小值.
y=a/x=6.76/3.9=5.2/3=1.7333333333
a''=(0.4x-0.18)/(x-0.45)^2-2(0.2x^2-0.18x-2.34)/(x-0.45)^3
=[(0.4x-0.18)(x-0.45)-2(0.2x^2-0.18x-2.34)]/(x-0.45)^3
=[0.4x^2-0.18x-0.18x+0.081-0.4x^2+0.36x+4.68]/(x-0.45)^3
=4.761/(x-0.45)^3>0,向上凹,有最小值.