To solve the problem of finding the minimum perimeter of a rectangle with an area of 100 square units, follow these steps:

Step 1: Define Variables

Let the length and width of the rectangle be (a) and (b) respectively.
Given: Area (A = a \times b = 100).
Perimeter (P = 2(a + b)).

Step 2: Apply AM-GM Inequality

The Arithmetic Mean-Geometric Mean (AM-GM) inequality states that for non-negative numbers (a) and (b):
[ \frac{a + b}{2} \geq \sqrt{ab} ]

Substitute (ab = 100):
[ \frac{a + b}{2} \geq \sqrt{100} = 10 ]
[ a + b \geq 20 ]

Step 3: Calculate Minimum Perimeter

Multiply both sides by 2 to get the perimeter:
[ P = 2(a + b) \geq 2 \times 20 = 40 ]

Equality holds when (a = b = 10) (since AM-GM equality requires (a = b)).

Answer: (\boxed{40})

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