My husband is building me a raised bed vegetable garden (google square foot gardening if you're intersted). Anyway ... if he uses 8 foot boards and builds it 4 feet by 4 feet we have 16 square feet. If he uses the SAME materials, but builds it 3 feet by 5 feet, we have 15 square feet. Am I just having a blond moment? Why does it work that way, why do we lose a foot even though we are using the same materials??

Because 2 shapes can have the same perimeter but different areas. With your 16 linear feet of perimeter, some options include: 1x7, area 7 2x6, area 12 3x5, area 15 4x4, area 16. The square will always maximize the area.

lets assume each foot holds one plant.. so a 3 X 5 hold 3 plants one direction and 5 plants another direction.. start with the 4 by 4. 4 plans long and 4 plants wide. x x x x x x x x x x x x x x x x so to go to a 3 x 5 you need to remove the 4th column (so you have 3 columns). x x x x x x x x x x x x x x x x To get 5 rows, you need one of each of the removed column and put it on each remaining column. x x x x x x x x x x x x x x x x This leaves the one remaining plant out in the open.. and is hence lost. That help?

A=lw 4'x4' = 16 square feet 3'x5' = 15 square feet A square will always maximize the area for a given perimeter in a rectangle. The more different the side lengths are, the smaller will be the area for the same perimeter. 2'x6' = 12 square feet 1'x7' = 7 square feet

if you have a constraint for the perimeter 2W+2L=C and you want to maximize the area A=L*W....you can use a little calculus from 2W+2L=C we can get W=(C-2L)/2 plug into A=L*W so A=L*(C-2L)/2=(1/2)(C*L)-L^2 findthe derivative of A with respect to L A'=C/2-2L and set equal to 0 and solve for L C/2=2L L=C/4 then W=(C-2L)/2=(C-2C/4)/2=(C/2)/2=C/4 so L=W and this is a max since A''=-2<0 for any value of L (2nd derivative test) you could have used Lagrange Multiplies if you wanted to but that would have been a little overkill

HMM, I'm afraid I could feel ku_alum's eyes go glassy all the way over on the Left Coast... Neatly explained, Catcherman.