Answer:
276 cm
460 cm if you require side lengths to be different
Step-by-step explanation:
You want the smallest possible perimeter of a rectangle with an area of 3²×23² cm².
Square
The rectangle will have the smallest perimeter when it is a square. The side lengths of the square will be the square root of the area:
s = √A
s = √(3²×23² cm²) = 3·23 cm = 69 cm
The perimeter of a square with side length 69 cm is ...
P = 4s
P = 4(69 cm) = 276 cm
The smallest possible perimeter of a rectangle is 276 cm.
Not a Square
Other possible rectangles with the given area and integer side lengths will have dimensions of ...
1 × 4761 ⇒ perimeter = 2(1 +4761) = 9524
3 × 1587 ⇒ perimeter = 2(3 +1587) = 3180
9 × 529 ⇒ perimeter = 2(9 +529) = 1076
23 × 207 ⇒ perimeter = 2(23 +207) = 460
Even though a square is a rectangle, you might interpret the problem as requiring the side lengths to be different. In that case ...
The smallest possible perimeter of a non-square rectangle is 460 cm.
