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