A mothball loses mass by evaporation at a rate proportional to its instantaneous surface area. If half the mass is lost in 100 days, how long will it be before the radius has decreased to one-half its initial value? How long before the mothball disappears completely? olution Assumption! "ass e#uals to volume, i.e. M
=
V
.
Solution . $
ince the surface area of a sphere of radius r is %π r , the first sentence says that dV dt
=
−
k %π r
$
%
. ince the volume of a sphere is
differentiation!
dV dr
=
& π r
&
%π r $ . 'he chain rule says that dV dt
=
−
k %π r
$
and
dV dt
, I can compute
dV
dV dr =
dt
dV dr =
dr dt
dr dt
%π r $
=
=
dr dt
dV dr
%π r $
by
dr dt
, so I have
.
'herefore −
k %π r
$
dr
%π r $
=
dt
dr
so
dt
=
−
k .
'his is an easy e#uation to solve! r = −kt + c . (rom the initial condition )at t*0+,
