The volume of a cube increases at a constant rate. Prove that the increase in its surface area varies inversely as the length of the side.
Given: a cube with volume increasing at a constant rate
To prove: the relation between the increase int eh surface area with the length of the side is inversely proportional.
Explanation: Let ‘a’ the length of the side of the cube.
Take the volume of the cube ‘V’
Then
As mentioned in the question, the rate of volume increase is constant, then
After substituting in the above equation, the values from equation (i) we get
Differentiating the equation with respect to t,
Take S as the surface area of the cube, then
After differentiating the surface area with respect to t, we get
Applying the derivatives, we get
After substituting the value from equation (ii) in the given equation we get
After taking out the like terms,
Now converting it to proportional, we get
Therefore, the relation between the length and the side of the cube is inversely proportional in the given condition.
Hence Proved.