Optimizing Area: Finding the Minimum Value of Sum of Squares for Cut Rectangles

[matpo39]

I am having a little bit of trouble with one of my math problems.

a) A rectanglewith length L and width W is cut into four smaller rectangles by two lines parallel to the sides. Find the minimum value of the sum of the squares of the areas of the smaller rectangles.

b) Show that the maximum of the sum of the squares of the areas occurs when cutting lines correspond to sides of the rectangle (so that there is only one rectangle).

i started part a) and this is what i got so far:

A=LW , A(small)=a , a=1/4*LW

so da/dL = (1/4)*W and da/dW= (1/4)*L

and the sum of the squares of all these are equal to 0 so
4[1/16(L^2+W^2)] = 0
1/4(L^2+W^2) = 0

I don't think that this is right though, can anyone help me out here?

thanks

 

[matpo39]

after thinking about it for a bit, i am now wondering if it would be easier if i were to try to use lagrange multipliers to solve this, but I am not too sure how i would implement them.

 

[Justin Lazear]

A=LW , A(small)=a , a=1/4*LW

Where're you getting this statement? Particularly that the area of the smaller rectangles is simply 1/4 the total area?

Let l be the length to the cut along the L side and w the length to the cut along the W side. Then,

a1 = lw
a2 = l(W-w)
a3 = (L - l)w
a4 = (L - l)(W - w)

and a1 + a2 + a3 + a4 = A (Note: this last equation won't help you).

See if you can't get that to work.

--Justin

 

[bizich]

I'm stuck on a similar problem and I'm not sure where to go from here. Help? Thanks a bunch!