Textbook Problem

The length ℓ, width w, and height h of a box change with time. At a certain instant the dimensions are ℓ = 1 m and w = h = 2 m, and ℓ and w are increasing at a rate of 2 m/s while h is decreasing at a rate of 3 m/s. At that instant find the rates at which the following quantities are changing.

(a) The volume

(b) The surface area

(c) The length of a diagonal