Pete has \(180\) cm of wire. He bends it to form the outline of his brand logo, which consists of a semicircle centered on top of a rectangle as shown in the diagram below. The width and length of the rectangle are \(3a\) cm and \(b\) cm respectively. The diameter of the semicircle is one-third the width of the rectangle.
Find the maximum possible area enclosed by the wire, showing that it is a maximum value. Give your answer correct to \(2\) decimal places.[6]
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