等腰梯形,面积为A,问底和腰各为多少是,两腰和底长之和最小?

Question

等腰梯形,面积为A,问底和腰各为多少是,两腰和底长之和最小?

Answer 1

设上底a,下底b,高h
A=1/2(a+b)*h,(a+b)*h=2A,a=2A/h-b,b-a=2b-2A/h,(b-a)/2=b-A/h
腰 c=√{h²+[(b-a)/2]²}
L=b+2c
=b+2√{h²+[(b-a)/2]²}
=b+2√[h²+(b-A/h)²]
=b+2√[h²+(b-A/h)²]