Saturation Math Example 4

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Example 4

medium
A bacteria culture saturates at 10610^6 cells. At time t=0t=0 there are 10410^4 cells. Write a logistic model P(t)P(t) with growth rate k=0.3k=0.3 per hour and find P(10)P(10).

Solution

  1. 1
    Standard form: P(t)=L1+Ae−ktP(t) = \frac{L}{1+Ae^{-kt}}. With L=106L=10^6, k=0.3k=0.3, and P(0)=104P(0)=10^4: 104=1061+A⇒1+A=100⇒A=9910^4=\frac{10^6}{1+A} \Rightarrow 1+A=100 \Rightarrow A=99.
  2. 2
    P(t)=1061+99e−0.3tP(t)=\frac{10^6}{1+99e^{-0.3t}}. Evaluate: P(10)=1061+99e−3=1061+99⋅0.04979=1061+4.929≈1065.929≈168,700P(10)=\frac{10^6}{1+99e^{-3}}=\frac{10^6}{1+99\cdot0.04979}=\frac{10^6}{1+4.929}\approx\frac{10^6}{5.929}\approx168{,}700.

Answer

P(t)=1061+99e−0.3tP(t)=\dfrac{10^6}{1+99e^{-0.3t}}; P(10)≈168,700P(10)\approx168{,}700
To find the constant AA, use the initial condition P(0)=P0P(0)=P_0: A=(L−P0)/P0=99A=(L-P_0)/P_0=99. The culture grows from 10410^4 toward the saturation limit 10610^6, reaching about 17%17\% of capacity by t=10t=10 hours.

About Saturation

Saturation is the phenomenon where a growing quantity approaches a limiting value asymptotically, with the rate of growth decreasing as the limit is approached.

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More Saturation Examples

Example 1 medium

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Example 2 hard

Identify the four parameters [formula], [formula], [formula] in [formula] and describe the qualitati

Example 3 easy

For [formula], state the saturation level and the value at [formula]. What does the saturation repre

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