Congratulations to Evan! His solution has been, once again, highlighted as an example solution on the Math Forum web site. Here is his solution:
Farmer Mead can build 9 rectangular pens that use all 36 meters. The amount of frogs that each pen can hold if each frog needs 1 square meter is equal to the area of the pen (area=length x width) The maximum number of frogs that can be in any pen is 81 frogs because the largest pen in 9mx9m - proving that a square is a rectangle, giving the most space.
To create at least 4 rectangular pens that all used 36 meters of fencing, first I picked lengths for each side that added up to 36 - remembering that rectangles had 2 sets of equal sides.
Side a and b were equal and side c and d were equal, so using that example,
a=17m and b=17m, c=1m and d=1m - adding all side = 36m. (17x2 + 1x2 = 36) . So my first pen was 17mx1m. I tried other combinations with the same strategy, and picked 2mx16m because 2x2+16x2=36. Next I tried 3mx15m because 3x2+15x2=36. I realized that this would work as long I kept decreasing the long side by 1 and increasing the short side by 1. Therefore, pen #4, would be 4mx14m, the fifth pen option would be 5mx13m, pen six would be 6mx12m, pen 7 would be created by using 7mx11m, pen 8 would be 8mx10m and the last pen would be 9mx9m.
If each frog needs one square meter of area, then according the area formula of length x width, my pens can hold the following number of frogs:
pen 1 can hold 17 frogs (17m x 1m=17m2)
pen 2 can hold 32 frogs (16m x 2m=32m2)
pen 3 can hold 45 frogs (15m x 3m =45m2)
pen 4 can hold 56 frogs (14m x 4m=56m2)
pen 5 can hold 65 frogs (13m x 5m=65m2)
pen 6 can hold 72 frogs (12m x 6m=72m2)
pen 7 can hold 77 frogs (11m x 7m=77m2)
pen 8 can hold 80 frogs (10m x 8m=80m2)
pen 9 can hold 81 frogs (9mx9m=81m2)
Extra: As you can see above the pen that holds the largest number of frogs is the pen that is 9m x 9m, pen #9. Since the pen has to be a rectangle, which means that 2 of the sides must be equal and parallel and the other 2 sides must be equal and parallel, and a pen that is 9m x 9m is a square, then a square must also be a rectangle. The rectangle that will be the largest is one that has equal sides all around. This pen would hold 81 frogs because 9m x 9m =81m2 = so this pen would hold 81 frogs.