| 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.
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