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A solid cube changes its volume such that its shape remains unchanged. For such a cube of unit volume, what will be the value of rate of change of volume?

A solid cube changes its volume such that its shape remains unchanged. For such a cube of unit volume, what will be the value of rate of change of volume? Correct Answer 3/2*(rate of change of area of any face of the cube)

Let x be the length of a side of the cube. If v be the volume and s the area of any face of the cube, then v = x3 and s = x2 Thus, dv/dt = dx3/dt = 3x2 (dx/dt) And ds/dt = dx2/dt = 2x(dx/dt) Now, (dv/dt)/(ds/dt) = 3x/2 Or, dv/dt = (3x/2)(ds/dt) Now, for a cube of unit volume we have, v = 1 =>x = 1 Therefore, for a cube of unit volume , we get, dv/dt = (3/2)(ds/dt) Thus the rate of change of volume = 3/2*(rate of change of area of any face of the cube)

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