Question

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) 3/8*(rate of change of area of any face of the cube)

(b) 3/4*(rate of change of area of any face of the cube)

(c) 3/10*(rate of change of area of any face of the cube)

(d) 3/2*(rate of change of area of any face of the cube)

The question was posed to me in an interview for internship.

The doubt is from Application of Derivative topic in section Application of Derivatives of Mathematics – Class 12

Answer 1

Right choice is (d) 3/2*(rate of change of area of any face of the cube)

Explanation: 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 = x^3 and s = x^2

Thus, dv/dt = dx^3/dt = 3x^2 (dx/dt)

And ds/dt = dx^2/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 [as, x is real]

Therefore, for a cube of unit volume [i.e. for x = 1], 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)