A square is inscribed in a circle. How fast is the area of the square changing when the area of the circle is increasing at the rate of 1 inch squared per minute?

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Answer 1 · 2016-11-02T20:11:47

#pi/4# square inches per minute.

Let #s# be the length of the sides of the square.

The area of the square (call it #A_S#) is #A_S = s^2#

The the radius of the inscribed circle is #s/2#.

The area of the circle, #A_C# is #pi(s/2)^2 = pi/4s^2#

We are interested in #(dA_C)/dt# at a moment when #(dA_S)/dt = 1#.

The variables #A_C# and #A_S# are related by

#(dA_C)/dt = pi/4 (dA_S)/dt #

We are told #(dA_S)/dt = 1#.

So we can conclude that #(dA_C)/dt = pi/4 #.