Expert: Paul Klarreich - 5/10/2007

Question
When a sperical mint is sucked, a simple model gives the rate of decrease of its radius as inversely proportional to the square of the radius. Initially the radius of the mint is 5mm and after 5 minutes the radius is 4mm.
Using this model find the radius of the mint 8 minutes after being put in the mouth and find what time the mint dissolves completely.   Many thanks

Answer
Questioner:   Mo
Category:  Calculus
Question:  When a spherical mint is sucked, a simple model gives the rate of decrease of its radius as inversely proportional to the square of the radius. Initially the radius of the mint is 5mm and after 5 minutes the radius is 4mm.
Using this model find the radius of the mint 8 minutes after being put in the mouth and find what time the mint dissolves completely.   Many thanks
.................................................
Hi, Mo,

Rate of decrease is  dr/dt.
Inversely proportional to square of radius means  -k/r^2

dr      k
-- = - ---
dt     r^2

Separate the variables:

r^2 dr = - k dt

Integrate:

r^3/3 = - kt + c  << small c.

r^3 = - 3kt + C   << Big C now.
Initial condition:  r = 5 at t = 0:

5^3 = -3k(0) + C

C = 5^3 = 125

r^3 = - 3kt + 125

Secondary condition:  r = 4  at  t = 5

64 = - 3k(5) + 125

64 = - 15k + 125

15k = 61

k = 61/15.  

r^3 = - 3(61/15)t + 125

r^3 = - (61/5)t + 125

Calculation: Find t when  r = 0. [Mint completely eaten.]

0^3 = - (61/5)t + 125

0 = - (61/5)t + 125

(61/5)t = 125

t = 625/61, or about 10 minutes.