Expert: Josh - 4/12/2007

Question
If V is the volume of a cube and X the length of an edge
Express dV/dT in terms of dX/dT. What is dV/dT when X is 5 and dX/dT=2.
(this is a grade 12 calculus question, which i had to spend 1 week finding)

Answer
Hello,

Assuming that T (let me write "t" instead) represents the time variable, the fact that dX/dt (the first derivative of X) is a constant tells us that the dimensions of the cube changes uniformly with time. i.e., both the length, width and height of the cube expands at a constant rate of 2 units per second.

If V represents the volume of the cube, then V(t)=X(t)^3. The reason I have written V as V(t) is to show that V changes as a function of time.

Now, using the chain rule of differentiation,
dV/dX= (dV/dt)*(dt/dX).
Since dX/dt is a constant, its inverse is simply dt/dX.

Thus, dV/dt=(dV/dX)*(dX/dt).......[1]

As dX/dt is given, we only have to figure out dV/dX.

From V=X^3, direct differentiation yields dV/dX= 3*X^2.
Expression [1] becomes dV/dt=3*X^2*(dX/dt).
When X=5, substituting dX/dt=2, we get dV/dt=3*(5^2)*2

Hope it makes more sense now:)