# Differential equation with sphere

## Homework Statement

A spherical mothball of original radius 1.2 cm slowly
evaporates such that 180 days later, its radius is only 1 cm.
Physically, the rate of evaporation dr/dt is proportional to the
surface area of the sphere. Determine a) the time required for
the radius of a new mothball to shrink to 25 percent of its
original radius, and b) the time required for the volume of a
new mothball to become half of its original value.

SA= 4 pi r^2
r(0)=1.2
r(180)=1

## The Attempt at a Solution

I started by saying -dr/dt = c1 4 pi r^2

where c1 is a constant of proportionality

Then through separation of variables I found that

1/r = 4 pi c1 t + c2

after imposing the initial conditions I found c1=7.383*10^-5 and c2=.833

so I have

1/r = 4 pi (7.383*10^-5) t + .833

and this gives me answers of
1) 2694 yr
2) 233.82 yr

but these aren't right. Any ideas of where I went wrong?

## Answers and Replies

mfb
Mentor
Your equation give the right values for t=0 and t=180, so I think you have some calculation error in the final steps.

I seriously doubt that mothballs evaporate like that. It would mean that evaporation is proportional to the square of the surface area.

Yeah, I think I might have the wrong initial equation, but I'm not sure of what it could be

Dick
Science Advisor
Homework Helper

## Homework Statement

A spherical mothball of original radius 1.2 cm slowly
evaporates such that 180 days later, its radius is only 1 cm.
Physically, the rate of evaporation dr/dt is proportional to the
surface area of the sphere. Determine a) the time required for
the radius of a new mothball to shrink to 25 percent of its
original radius, and b) the time required for the volume of a
new mothball to become half of its original value.

SA= 4 pi r^2
r(0)=1.2
r(180)=1

## The Attempt at a Solution

I started by saying -dr/dt = c1 4 pi r^2

where c1 is a constant of proportionality

Then through separation of variables I found that

1/r = 4 pi c1 t + c2

after imposing the initial conditions I found c1=7.383*10^-5 and c2=.833

so I have

1/r = 4 pi (7.383*10^-5) t + .833

and this gives me answers of
1) 2694 yr
2) 233.82 yr

but these aren't right. Any ideas of where I went wrong?

1/r = 4 pi (7.383*10^-5) t + .833 looks ok. Why are you giving the times in yrs? Don't you mean days? And I'd check those again.

1/r = 4 pi (7.383*10^-5) t + .833 looks ok. Why are you giving the times in yrs? Don't you mean days? And I'd check those again.

Wow. Don't I feel silly. I was working on several problems at once and I guess I forgot this one was in days, not years and it worked. Good eye! Also, Thanks a ton!