A thin metallic wire in the shape of a circular ring has its enclosed area increasing at a uniform rate when heated. Show that the rate of change of circumference varies inversely as the radius.
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Whenever a question asks to show that a quantity $Y$ varies inversely as $X$, always look to find an equation of the form $Y = \frac{\text{Constant}}{X}$. Here, since the area rate is constant, the expansion speed of the boundary slows down as the circle grows wider because the same amount of added area must be spread over a larger perimeter.