(a) The side of the cube is of length x cm.
The volume of the cube is given by width x height x depth, V = x^3.
A cube has 6 faces. The area of each face is given by height x width (x^2).
Total surface area = S = 6x^2.
Differentiating the surface area with respect to the length of each side gives dS/dx = 12x.
The question states that the surface area of the cube is increasing by 8cm per second, so the rate of change of the surface area with respect to time is dS/dt = 8.
Now, using the chain rule:
dx/dt = dx/dS x dS/dt
dx/dt = [1/12x][8]
dx/dt = 2/3x
[k]
(b) S = 6x^2
Rearranging to get x in terms of S gives x = (S/6)^1/2
V = x^3
Substituting x in terms of S into the volume gives
V = (S/6)^3/2
Raising both sides to the power of 2/3 gives
V^2/3 = S/6
6V^2/3 = S
6(2/3)V^-1/3 = dS/dV
4V^-1/3 = dS/dV
dV/dt = dV/dS x dS/dt
dV/dt = [V^(1/3)/4][8]
dV/dt = 2V^1/3
I really hope that's helpful. Was a very tricky question for me, got stuck on it today when doing this past paper.