jmc

Question

A fenced, rectangular field measures 24 meters by 52 meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the field. What is the largest number of square test plots into which the field can be partitioned using all or some of the 1994 meters of fence?

Accepted Answer

Suppose there are n squares in every column of the grid, so there are 5224n = 136n squares in every row. Then 6|n, and our goal is to maximize the value of n. Each vertical fence has length 24, and there are 136n - 1 vertical fences; each horizontal fence has length 52, and there are n-1 such fences. Then the total length of the internal fencing is 24(13n6-1) + 52(n-1) = 104n - 76 1994 n 103552 19.9, so n 19. The largest multiple of 6 that is 19 is n = 18, which we can easily verify works, and the answer is 136n^2 = 702.