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Growth of Mold proportional to the amount present
 

Moldy Growth

A mold grows at a rate proportional to the amount present. Initially, its weight is 3 grams. After 2 days, its weight is 5 grams. How much does it weigh after 12 days?



Problem Moderated by: MrT

  
 
 
 

Problem Solution

Since the rate of growth is equal to dm/dt, (where m is the mass of the mold), then we get the differential equation

dm/dt = km,

where k is the constant of proportionality. Separating the variables gives the equation

dm/m = kdt.

Integrating both sides gives

ln m = kt + C

Changing to exponential form gives

m = ekt + C

which is equivalent to

m = C1ekt

Since m = 3 when t = 0, C1 = 3.

Substituting m = 5 when t = 2 gives the following:

5 = 3e2k

Solving this gives (ln(5/3))/2 = k, or

k = .2554128. . .

Now, evaluate the equation when t = 12 and get

m = 64.3 grams.

2003 Archive

  
 
 



 
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