Yahoo Answers: Answers and Comments for Math Homework? [Mathematics]
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Sat, 18 Jan 2020 03:41:21 +0000
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Yahoo Answers: Answers and Comments for Math Homework? [Mathematics]
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From Captain Matticus, LandPiratesInc: P(15) = 300
P(40) = 1700
P(t) = a * 2^(t/k)
...
https://ca.answers.yahoo.com/question/index?qid=20200118034121AAyr3PM
https://ca.answers.yahoo.com/question/index?qid=20200118034121AAyr3PM
Sat, 18 Jan 2020 03:59:07 +0000
P(15) = 300
P(40) = 1700
P(t) = a * 2^(t/k)
k is the doubling time
a is the initial population
a * 2^(40/k) = 1700
a * 2^(15/k) = 300
a * 2^(40/k) / (a * 2^(15/k)) = 1700/300
2^(40/k  15/k) = 17/3
2^(25/k) = 17/3
2^(1/k) = (17/3)^(1/25)
P(t) = a * 2^(t/k)
P(t) = a * (2^(1/k))^t
P(t) = a * ((17/3)^(1/25))^t
P(t) = a * (17/3)^(t/25)
Isn't that lovely? You can get the same effect by starting out with e, or 10, or a googol, or whatever. It doesn't have to be a 2, but I felt that by using a 2, you could see why the population doubles every k minutes. But it has served its purpose and now we have something much nicer to look at. Now, let's solve for a.
P(15) = 300
300 = a * (17/3)^(15/25)
300 = a * (17/3)^(3/5)
300 / (17/3)^(3/5) = a
a = 300 * (17/3)^(3/5)
a = 105.9557804410011235032485352032...
Round to the nearest whole value.
a = 106
There were 106 bacteria initially.
2a = a * (17/3)^(t/25)
2 = (17/3)^(t/25)
2^25 = (17/3)^t
25 * ln(2) = t * ln(17/3)
25 * ln(2) / ln(17/3) = t
t = 9.990008630613594640745895492379...
9.99 minutes is the doubling time
P(85) = 106 * (17/3)^(85/25)
P(85) = 106 * (17/3)^(17/5)
P(85) = 106 * (17/3)^(3.4)
P(85) = 38,602.900010311218626673704375427....
Or more accurately
P(85) = 300 * (17/3)^(3/5) * (17/3)^(17/5)
P(85) = 300 * (17/3)^(14/5)
P(85) = 300 * (17/3)^(2.8)
P(85) = 38,586.796206400523465759934256926
Take your pick, round accordingly. I was always taught to save all of my rounding until the very end, but that's just me.
14000 = 106 * (17/3)^(t/25)
14000/106 = (17/3)^(t/25)
(14000/106)^(25) = (17/3)^t
25 * ln(14000/106) = t * ln(17/3)
t = 25 * ln(14000/106) / ln(17/3)
Or....
14000 = 300 * (17/3)^(3/5) * (17/3)^(t/25)
140/3 = (17/3)^(t/25  15/25)
(140/3) = (17/3)^((t  15) / 25)
(140/3)^(25) = (17/3)^(t  15)
25 * ln(140/3) = (t  15) * ln(17/3)
25 * ln(140/3) + 15 * ln(17/3) = ln(17/3) * t
t = (25 * ln(140/3) + 15 * ln(17/3)) / ln(17/3)
Wolfram Alpha can evaluate those for you, if you copy and paste them exactly.