As a raindrop falls, it evaporates while retaining its spherical shape. If we make the further assumptions that the rate at which the raindrop evaporates is proportional to its surface area and that air resistance is negligible, then a model for the velocityv(t)

of the raindrop is

dv

dt

+3(k/)

(k/)t+r₀

v=g.

Hereis the density of water,r₀

is the radius of the raindrop att= 0,k< 0

is the constant of proportionality, and the downward direction is taken to be the positive direction.

(a) Solve forv(t)

if the raindrop falls from rest.

v(t) =

(b) This model assumes that the rate at which the raindrop evaporatesthat is, the rate at which it loses massis proportional to its surface area, with constant of proportionalityk< 0.

This assumption implies that the rate at which the radiusrof the raindrop decreases is a constant. Show that the radius of the raindrop at timetisr(t) = (k/)t+r₀.

LettingAdenote the surface area of the raindrop andmthe mass, the model assumes that

dm

dt

=kA.

Since mass equals density times

---Select---

force

velocity

work

volume

area

and a sphere of radiusrhas volumeand surface area, we obtain the following formulas formandAin terms ofandr.

m=A=

Plugging these formulas into the differential equation gives:

d

dt

=k4r²dr

dt

=k4r²dr

dt

=k(assumingr0)dr

dt

=.

Integrating the above differential equation gives

r(t) =t+C.

Plugging inr(0) =r₀

givesC=,

so that

r(t) =.

(c) Ifr₀=0.03ft andr=0.005ft

10 seconds after the raindrop falls from a cloud, determine the time at which the raindrop has evaporated completely. (Round your answer to one decimal place.)

sec