请用拉格朗日乘数法求如下目标函数的极值find the minimal area of a pyramid with rectangular base of edges of length a, b and heigh h>0 if the volume of the pyramid is 1也可以用matlab 算出写出答案即可 急急急急急急急
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![请用拉格朗日乘数法求如下目标函数的极值find the minimal area of a pyramid with rectangular base of edges of length a, b and heigh h>0 if the volume of the pyramid is 1也可以用matlab 算出写出答案即可 急急急急急急急](/uploads/image/z/8502800-32-0.jpg?t=%E8%AF%B7%E7%94%A8%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E4%B9%98%E6%95%B0%E6%B3%95%E6%B1%82%E5%A6%82%E4%B8%8B%E7%9B%AE%E6%A0%87%E5%87%BD%E6%95%B0%E7%9A%84%E6%9E%81%E5%80%BCfind+the+minimal+area+of+a+pyramid+with+rectangular+base+of+edges+of+length+a%2C+b+and+heigh+h%3E0+if+the+volume+of+the+pyramid+is+1%E4%B9%9F%E5%8F%AF%E4%BB%A5%E7%94%A8matlab+%E7%AE%97%E5%87%BA%E5%86%99%E5%87%BA%E7%AD%94%E6%A1%88%E5%8D%B3%E5%8F%AF+%E6%80%A5%E6%80%A5%E6%80%A5%E6%80%A5%E6%80%A5%E6%80%A5%E6%80%A5)
请用拉格朗日乘数法求如下目标函数的极值find the minimal area of a pyramid with rectangular base of edges of length a, b and heigh h>0 if the volume of the pyramid is 1也可以用matlab 算出写出答案即可 急急急急急急急
请用拉格朗日乘数法求如下目标函数的极值
find the minimal area of a pyramid with rectangular base of edges of length a, b and heigh h>0 if the volume of the pyramid is 1
也可以用matlab 算出写出答案即可 急急急急急急急!!!!!!!!!!
谢谢
请用拉格朗日乘数法求如下目标函数的极值find the minimal area of a pyramid with rectangular base of edges of length a, b and heigh h>0 if the volume of the pyramid is 1也可以用matlab 算出写出答案即可 急急急急急急急
这是求以长a宽b的矩形为底面,高h的四棱锥的面积最小
体积 1/3 abh=1,所以 abh=3
顶点在底面的投影是底面的中心
侧面的三角形的高,及相应面积为
h1=sqrt(h^2+(a/2)^2),S1= 1/2 b*h1
h2=sqrt(h^2+(b/2)^2),S2=1/2 a*h2
总面积为
S=ab+b*sqrt(h^2+(a/2)^2)+a*sqrt(h^2+(b/2)^2),
满足 abh=3
拉格朗日乘数法
设f=S-c*(abh-3)
分别求偏导数,
a=b=(3/2*根2)^(1/3)=3^(1/3)/(2^(1/6))
h=6^(1/3)
此时 面积最小值为 6.6039