Question

Find the dimensions of the rectangular box with largest volume if the total surface area is given as 64 cm2

Answer 1

The problem is to optimize (find the maximum The problem is to maximiz the function 

f(x,y,z)=xyz 
With the constrain 
2(xy + xz + yz)=64; xy+xz+yz=32 
Using the Lagrange Multipliers 
F(x,y,z) = xyz - £(xy+xz +yz-32) 
Deriving with respect to x: 
yz - £(y+z)=0 ....i
Deriving with respect to y: 
xz - £(x+z)=0 ...ii
Deriving with respect to z: 
xy - 2£(x+y)=0 ....iii
Deriving with respect to £: 
xy+xz+yz=32 .....iv
From (i) and (ii) 
yz/2(y+z) = xz/2(x+z) 
y/(y+z) = x/(x+z) 
yx+yz=xy+xz 
y=x 
From (i) and (iii) 
x=z 
So, from (iv) 
x^2+x^2+x^2=32 
x^2=32/3 
x=y=z=sqrt (32/3) 
Vmax = sqrt (32/3)^3