SOLUTION: A bacteria culture starts with 160 bacteria and grows at a rate proportional to its size. After 3 hours there will be 480 bacteria. (a) Express the population P after t hours as

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Question 1179719: A bacteria culture starts with 160 bacteria and grows at a rate proportional to its size. After 3 hours there will be 480 bacteria.
(a) Express the population P after t hours as a function of t.
P(t)=

(b) What will be the population after 4 hours?

(c) How long will it take for the population to reach 2770? Round your answer to two decimal places.
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
A bacteria culture starts with 160 bacteria and grows at a rate proportional to its size.
After 3 hours there will be 480 bacteria.
(a) Express the population P after t hours as a function of t.
P%28t%29=+p%2A2%5E%28tk%29, where
P(t) = 480
p = 160
t = 3
k = constant
160%2A2%5E%283k%29+=+480
2%5E%283k%29+=+480%2F160
2%5E%283k%29+=+3
using logs
3k*ln(2) = ln(3)
3k = ln%2830%29%2Fln%282%29
3k = 1.585
k = 1.585%2F3
k = .528
The equation
P%28t%29+=+p%2A2%5E%28.528t%29
:
(b) What will be the population after 4 hours?
P%28t%29+=+160%2A2%5E%28.528%2A4%29
p(t) = 692
:
(c) How long will it take for the population to reach 2770?
Round your answer to two decimal places.
160%2A2%5E%28.528t%29+=+2770
2%5E%28.528t%29+=+2770%2F160
2%5E%28.528t%29+=+17.3125
.528t = ln%2817.315%29%2Fln%282%29
.528t = 4.1137
t = 4.1137%2F.528
t = 7.79 hrs