Question

A square region with perimeter 60 inches is made with square inch tiles. Bob removes one tile from the square and rearranges the remaining tiles without any overlap to make a rectangular region with minimum perimeter. How many inches are in the perimeter

Answer 1

60 inches

Explanation:

The perimeter will be minimized when the tiles are rearranged to form a rectangle with dimensions as close to each other as possible.

Rectangle dimensions

The square has a side length of ...

P = 4s ⇒ s = P/4 . . . . . P is perimeter, s is side length

(60 in)/4 = 15 in . . . . length of the side of the square

The square has an area of ...

A = s²

A = (15 in)² = 225 in²

When 1 tile is removed, the resulting area, in square inches, is ...

A = 225 -1 = 15² -1² = (15 -1)(15 +1) = 14×16

Perimeter

The perimeter of a rectangle is given by ...

P = 2(L+W)

For this rectangle, the perimeter is ...

P = 2(14 in + 16 in) = 2(30 in) = 60 in

The perimeter of the rectangle is 60 inches.