Question

A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was 1350 bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after 3 hours?

Answer 1

Answer:

• p(t) = 1350·2^(t/20); p(180) = 691,200
• p(t) ≈ 1350·e^(0.034657t); p(180) ≈ 691,155

Step-by-step explanation:

The situation can be represented exactly by the exponential equation ...

  p = 1350·2^(t/20) . . . . . . t in minutes; p is population count

After 3 hours (180 minutes), the population is ...

  p = 1350·2^(180/20) = 1350·2^9 = 691,200 . . . population after 3 hours

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Alternate Solution

Some prefer their exponential functions to be in the form ...

  p = a·e^(kt)

where "a" is the initial population count, and k is determined by the growth rate.

We can rewrite the two forms to allow us to match the various constants:

  p = 1350·(2^(1/20))^t = a·(e^k)^t

This shows us that a=1350 and e^k = 2^(1/20). Then k = ln(2)/20 ≈ 0.034657

Now, the exponential function can be written as ...

  p = 1350·e^(0.034657t)

Filling in t=180 in this equation, we find ...

  p = 1350·e^(0.034657·180) ≈ 1350·511.967 ≈ 691,155.33 ≈ 691,155

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Comment on the answers

My personal preference is for the exact form of the equation and the integer answer. I suppose your grader is expecting the second answer, since you were asked to round the equation to 5 significant digits, and to round the population to the nearest integer.