It is known that the circumference of the rectangle is 36cm, and the rectangle rotates around one of its sides to form a cylinder. When the length and width of the rectangle are respectively, the side area of rotation is the largest?

Let the length of rectangle be a, the width be B, the perimeter of ∵ rectangle be 36, and ∵ 2 (a + b) = 36. The solution is: B = 18-a, ∵ the side area of the cylinder formed by rotation is: 2 π AB, ∵ the maximum side area is required, that is, the maximum value of AB is calculated, ab = a (18-a) = 18a-a2 = - (A-9) 2 + 81, ∵ when a = 9, AB has the maximum value of 81, and B = 9. Answer: when the length and width of rectangle are both 9, the side area of the cylinder formed by rotation is the maximum

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