Suppose you consider a rectangular floor space of 36 square meters with whole-number side lengths. Which design has the least perimeter? Which has the greatest perimeter? Explain your reasoning.

If the length of the floor is x m, then its width is 36/x. The perimeter of the floor is p(x) = 2x + 72/x. Since both the length and the width take whole number values, x is a divisor of 36. The number 36 has the following divisors: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Consider the following table

x 1 2 3 4 6 9 12 18 36

p(x) 74 40 30 26 24 26 30 40 74

We see that the square design 6x6 has the least perimeter 24, whereas the design 1x36 has the greatest perimeter 74.