The minimum is when the rectangle is square, A = s^2. Since you have the area, you know what the side length is, and therefore what the perimeter is.
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Short answer: the minimum perimeter of a rectangle with a given area is always a square. Thus, the the minimum perimeter occurs when the side length, s, of each side of the rectangle satisfies:
s^2 = 500 ==> s = √500 = 10√5.
Thus, the minimum perimeter is 4(10√5) = 40√5.
You can also show this via Calculus. If x is the length of the rectangle and y is the width, then the area of the rectangle is xy. Since the area is 500, you have:
xy = 500.
The perimeter of the rectangle is P = 2x + 2y. Solve xy = 500 for any variable you wish and find the value of that variable that minimizes the area. You'll get the same answer as above.
s^2 = 500 ==> s = √500 = 10√5.
Thus, the minimum perimeter is 4(10√5) = 40√5.
You can also show this via Calculus. If x is the length of the rectangle and y is the width, then the area of the rectangle is xy. Since the area is 500, you have:
xy = 500.
The perimeter of the rectangle is P = 2x + 2y. Solve xy = 500 for any variable you wish and find the value of that variable that minimizes the area. You'll get the same answer as above.
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Answer is 4 * sqrt 500 cm
= 4 * 22.3606 = 89.4427 cm
= 4 * 22.3606 = 89.4427 cm