Answer:
a) C(t) = 3000e^(rt)
b) C(8) = 47400 yeast cells.
c) 13870 yeast cells a hour.
Step-by-step explanation:
The growth of the culture of yeast in hours can be modeled by the following differential equation:
1) dC/dt = rC,
where C is the number of cells and r is the growth rate.
For question a), to write an expression for the number of yeast cells after t hours, we need to solve the differential equation 1). I am going to solve it by the variable separation method.
dC/C = rdt
Integrating both sides, we have:
ln C = rt + C0
where C0 is the initial population of cells.
We need to isolate C in this equation, so we do this
e^(ln C) = e^(rt + C0)
So
C(t) = C0e^(rt)
The initial population of cells is given as 3000, so:
C(t) = 3000e^(rt)
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b)
After two hours, the number of cells grows. So C(2) = 6000. This is helpful so we can find the growth rate r.
6000 = 3000e^(2r)
e^(2r) = 2
ln(e^(2r)) = ln 2
2r = 0.69
r = 0.345
Now we have C(t) = 3000e^(0.345t), so
C(8) = 3000e^(0.345*8) = 47400 yeast cells.
c)
C(7) = 3000e^(0.345*7) = 33570 yeast cells.
C(8) - C(7) = 47400 - 33570 = 13870 yeast cells. So, at 8 hours, the population of yeast cells is increasing at the rate of 13870 yeast cells a hour.
