At what rate is the diagonal of a cube increasing if its edges are increasing at a rate of 2 cm/s

Question

This is a related rates problem

 

Answer 1

If you think about cube, there are two diagonals. One is a diagonal of a face(D_f) and the other is a diagonal of a cube(D_c). Let's x is a length of the edge.

Then, D_f = √(x² + x²) = x √2 ----------------------(1)

         D_c = √(x² + x² + x²) = x √3 ---------------(2)

Now, if you differentiate each equations for the time t,

         d(Df)/dt = dx/dt times √2   -------------------(3)

         d(Dc)/dt = dx/dt times √3 --------------------(4)

The edges are increasing at a rate of 2cm/s, this means that  dx/dt = 2 cm/s in equation (3) and (4).

Therefore, d(Df)/dt = 2 √2   [cm/s]

                and d(Dc)dt = 2√3  [cm/s]

I am not sure which diagonal does the problem want. Generally, the diagonal of a cube is Dc in here.

I hope this will help you.

 

 

Answer 2

 

For a cube of side 1, the outside (surface)  diagonals are 2^(1/2).  From Pythagoras, √(1^2 + 1^2) = (2)^(1/2).

The inside diagonal is found in a like manner, and is (3)^(1/2).  Since everything must enlarge uniformly, the inside diagonal  increases at (2 cm)(3)^(1/2), or 2√3 cm.  The surface diagonal increases at 2√2 cm.