i want to build a small rectangular gardening area at the back of my house with the perimeter 1500m of fencing.

Question

the fence is required along remaining 3 sides of the area.

what is the maximum area that can enclosed with 1500m of fencing?

Answer 1

P=2l+2w but one length is not used

P=l+2w

1500=l+2w

1500-2w=l

A=lw

A=(1500-2w)w

A=1500w-2w^2

this is a parabola with a maximum at (h,k) where h=-b/2a

-1500/-4=375 m

the maximum area is always a square

you have a rectangle with three sides

make the width 375 m and add 375 m to 375 m for the length to get 750 m

you have a rectangle 375 m by 750 m by 375 m by "the house"

the maximum area is:

A=lw

A=750*375

A=281,250 m²

notice that the rectangle is the same as two 375 by 375 squares next to each other