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Solution to Problem 10 in Review Sheet
A culture contains 1500 bacteria initially and doubles every 30 minutes.
1. Find a function that models the number of bacteria n(t) after t minutes.
2. Find the number of bacteria after 2 hours.
3. After how many minutes will the culture contain 4000 bacteria?
Solution.
1. n(t) = 1500ekt , where k is the growth rate. Let’s find k.
We’re given that n(30) = 3000. Thus, we have the equation
3000 = 1500ek·30 ⇒ 2 = e30k ⇒ ln 2 = ln e30k ⇒ ln 2 = 30k ⇒ k =
ln 2
Thus, n(t) = 1500e 30 t .
ln 2
.
30
ln 2
2. After 2 hours, t = 120 minutes. Thus, there will be n(120) = 1500e 30 ·120 = 1500e4 ln 2 =
4
1500eln 2 = 1500 · 24 = 24000 bacteria present.
ln 2
3. Find t such that n(t) = 4000. That is, we need to solve 1500e 30 t = 4000.
ln 2
ln 2
ln 3
We have 1500
= e 30 t ⇒ 38 = e 30 t ⇒ ln 38 = ln302 t ⇒ t = 30 ln 28 minutes.
4000