Mathematical Modeling Math Example 2

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Example 2

medium
A population of bacteria doubles every hour. If the initial count is P0=500P_0 = 500, write a model for the population P(t)P(t) after tt hours and find P(4)P(4).

Solution

  1. 1
    Each hour the population multiplies by 2, so after tt hours: P(t)=P0⋅2tP(t) = P_0 \cdot 2^t.
  2. 2
    Substitute P0=500P_0 = 500: P(t)=500⋅2tP(t) = 500 \cdot 2^t.
  3. 3
    For t=4t = 4: P(4)=500⋅24=500⋅16=8000P(4) = 500 \cdot 2^4 = 500 \cdot 16 = 8000.

Answer

P(t)=500⋅2t,P(4)=8000P(t) = 500 \cdot 2^t,\quad P(4) = 8000
Exponential models arise whenever a quantity grows by a fixed multiplicative factor per unit time. The key insight is that repeated multiplication by 2 corresponds to the function 2t2^t.

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The process of using mathematical structures — functions, equations, distributions — to represent, analyze, and predict real-world phenomena.

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