SOLUTION: A bacteria culture starts with 560 bacteria and grows at a rate proportional to its size. After 2 hours there will be 1120 bacteria.
(a) Express the population P after t hours a
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(a) Express the population P after t hours a
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Question 1153464: A bacteria culture starts with 560 bacteria and grows at a rate proportional to its size. After 2 hours there will be 1120 bacteria.
(a) Express the population P after t hours as a function of t.
P(t)=____
(b) What will be the population after 8 hours?
___ bacteria
(c) How long will it take for the population to reach 2490? Give your answer accurate to at least 2 decimal places.
___ hours Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website! 1120/560 = 2
the formula to use for this problem is:
p(t) = 560 * (1 + r) ^ t
p(t) becomes 1120 and t becomes 2 to get:
1120 = 560 * (1 + r) ^ 2
divide both sides by 560 to get:
1120 / 560 = (1 + r) ^ 2
take the square root of both sides to get:
(1120 / 560) ^ (1/2) = 1 + r
solve for 1 + r to get:
1 + r = 1.414215362
that's your hourly growth factor.
in 2 hours, you get 560 * 1.414215362 ^ 2 = 1120
in 8 hours, you get 560 * 1.414215362 ^ 8 = 8960
to find how long for the population to becomes 2490, you would do the following.
formula is p(t) = 560 * 1.414215362 ^ t
when p(t) is 2490, the formula becomes:
2490 = 560 * 1.414215362 ^ t
divide both sides by 560 to get
2490 / 560 = 1.414215362 ^ t
take the log of both sides of the equation to get:
log(2490 / 560) = log(1.414215362 ^ t)
since log(x^a) = a * log(x), your formula becomes:
log(2490 / 560) = t * log(1.414215362
divide both sides by log(1.414215362) to get:
log(2490 / 560) / log(1.414215362) = t
solve for t to get:
t = 4.30529402.
confirm by replacing t with that in the original equation to get:
p(t) = 560 * 1.414215362 ^ 4.30529402 = 2490
this confirms the solution is correct.
you can graph the equation as y = 560 * 1.414215362 ^ x
it would look like this:
the graphing software rounds the answer to 3 decimal places.
4.3052.... becomes 4.305.
