Q. 105.0( 2 Votes )

# The volume of a c

Answer :

Given: a volume of cube increasing at a constant rate

To prove: the increase in its surface area varies inversely as the length of the side

Explanation: Let the length of the side of the cube be ‘a’.

Let V be the volume of the cube,

Then V = a3……..(i)

As per the given criteria the volume is increasing at a uniform rate, then Now substituting the value from equation (i) in above equation, we get Now differentiating with respect to t we get  Now let S be the surface area of the cube, then

S = 6a2

Now differentiating surface area with respect to t, we get Applying the derivatives, we get Now substituting value from equation (ii) in the above equation we get Cancelling the like terms we get Converting this to proportionality, we get Hence the surface area of the cube with given condition varies inversely as the length of the side of the cube.

Hence Proved

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