A culture of yeast grows at a rate proportional to its size. If the initial population is $2000$ cells and it doubles after $3$ hours, answer the following questions.

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1. Write an expression for the number of yeast cells after $t$ hours.
Answer: $P(t) =$

2. Find the number of yeast cells after $6$ hours.
Answer:

3. Find the rate at which the population of yeast cells is increasing at $6$ hours.
Answer (in cells per hour):

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