Question #26b83
3 Answers
Explanation:
The rate of increase of the area of the surface can be expressed as:
We can find the rate of increase of the volume with the chain rule:
We must then express the volume as function of the surface.
As:
So, finally:
Rate of Change of volume is
Explanation:
Let us set up the following variables:
{(V, "Volume of the sphere",cm^3), (S, "Surface Area of the sphere",cm^2), (t, "time",sec) :}
The Volume of the sphere is given by the standard formula:
V=4/3pir^3
Differentiating wrt
(dV)/(dr) = 4pir^2
The Surface Area of the sphere is given by the standard formula:
S = 4pir^2
Differentiating wrt
(dS)/(dr) = 8pir
We are given that:
(dS)/dt = 8 \ cm^2s^-1
By the chain rule:
(dV)/dt = (dV)/(dr) * (dr)/(dS) * (dS)/(dt)
" " = (4pir^2) * (1/(8pir)) * (8)
" " = 4r \ \ cm^3s^(-1)