Applications of the Chain Rule 2
When the volume of a solid is increasing, so in general are the dimensions and surface area of the solid. If the solid maintans it shape, so that at each point in time it has a mathematically similar shape to the original, then we can use the chain rule to relate the rate of increase of volum to the rate of increase of the surface are or length of the solid.
Suppose that a cube is increasing in volume at the rate of 4 cm 3 /s. The cube has sides of length
so the volume of the cube is
and the surface area is
Suppose it is desired to find the rate at which the surface area is increasing when
We can use the Chain Rule, which states
and
From the first of these, with
so that when
then
cm/s
From the second of these, with
so that when
cm 2 /s.
Alternatively, we could have used the Chain Rule in the form
With
When
so
and
cm 2 /s.
