Rates of Change

Peng

2008-10-14 12:57 pm

The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function

n= _______a_______
　　1 + be ^(-0.7t)

where t is measured in hours. At ime t=0 the population is 20 cells and is increasing at a rate of 12 cells/hour. Find the values of a and b. According to this model, what happens to the yast population in the long run?

回答 (1)

魏王將張遼

2008-10-14 5:20 pm

✔ 最佳答案

Sub t = 0, we have n = 20, then
20 = a/(1 + b)
20 + 20b = a ... (1)
Then, taking differentiation of n w.r.t. t:
dn/dt = [a d(1 + be-0.7t)/dt]/(1 + be-0.7t)2
-0.7abe-0.7t/(1 + be-0.7t)2
Sub t = 0, we have:
-0.7ab/(1 + b)2 = 12 ... (2)
Sub (1) into (2):
-0.7(20 + 20b)b/(1 + b)2 = 12
-14b - 14b2 = 12 + 24b + 12b2
26b2 + 38b + 12 = 0
13b2 + 19b + 6 = 0
(13b + 6)(b + 1) = 0
b = -6/13 or -1 (rejected since the denominator of n will be zero when t = 0)
a = 140/13
So n can be expressed as:
n = (140/13)/(1 - 6e-0.7t/13)
= 140/(13 - 6e-0.7t)
So as t tends to infinity, the term e-0.7t vanishes and therefore n approaches 140/13.

參考: My Maths knowledge

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