- #1
phosgene
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- 1
Homework Statement
Maximise the volume of a rectangular prism with the following constraints: the surface area must equal 2m^2 and the total edge length must be 12m.
Homework Equations
Using Lagrange multipliers, we construct the function we want to optimise with
[itex]h(x,y,z, λ_{1}, λ_{2}) = f(x,y,z) + λ_{1}g_{1}(x,y,z) + λ_{2}g_{2}(x,y,z)[/itex]
The Attempt at a Solution
In this case our f(x,y,z) is the volume of the rectangular prism, so [itex]f(x,y,z)=xyz[/itex], where we take x to be the length, y the widge and z the depth. [itex]g_{1}(x,y,z)=λ_{1}(xy + yz + zx - 1)[/itex], the surface area constraint and [itex]g_{2}(x,y,z)=λ_{2}(x+y+z-3)[/itex] (the edge length constraint).
The function to be optimised is then [itex]h(x,y,z, λ_{1}, λ_{2}) = xyz + λ_{1}(xy + yz + zx - 1) + λ_{2}(x+y+z-3)[/itex]
Obtain all the partial derivatives:
1. [itex]\frac{∂h}{∂x} = yz + λ_{1}(y+z) + λ_{2} = 0[/itex]
2. [itex]\frac{∂h}{∂y} = xz + λ_{1}(x+z) + λ_{2} = 0[/itex]
3. [itex]\frac{∂h}{∂z} = yx + λ_{1}(y+z) + λ_{2} = 0[/itex]
4. [itex]\frac{∂h}{∂λ_{1}} = xy + yx + zy -1 = 0 [/itex]
5. [itex]\frac{∂h}{∂λ_{2}} = x + y + z -3 = 0 [/itex]
Add 1, 2 and 3. Then sub in 4 and 5 to obtain the result [itex]λ_{1}=-(3λ_{2}+1)/6[/itex]
But from here I'm totally lost. I can re-arrange 1,2 and 3 to get λ2 by itself, then equate them all to get
[itex]\frac{\frac{1}{6}(y+z) - yz}{1-\frac{1}{2}(y+z)}=\frac{\frac{1}{6}(y+x) - yx}{1-\frac{1}{2}(y+x)}=\frac{\frac{1}{6}(z+x) - zx}{1-\frac{1}{2}(z+x)}[/itex]
But doesn't this imply that x=y=z? This cannot satisfy both 4 *and* 5.