Answer:
Step-by-step explanation:
It often works well to write an exponential expression as ...
value = (initial value)×(growth factor)^(t/(growth period))
(a) Here, the growth factor for the bacteria is given as 230/100 = 2.3 in a period of 1 hour. The initial number is 100, so we can write the pupulation function as ...
P(t) = 100·2.3^t
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(b) P(2) = 100·2.3^2 = 529 . . . number after 2 hours
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(c) P'(t) = ln(2.3)P(t) ≈ 83.2909·2.3^t
P'(2) = 83.2909·2.3^2 ≈ 441 . . . bacteria per hour
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(d) We want to find t such that ...
P(t) = 10000
100·2.3^t = 10000 . . . substitute for P(t)
2.3^t = 100 . . . . . . . . divide by 100
t·log(2.3) = log(100)
t = 2/log(2.3) ≈ 5.5 . . . hours until the population reaches 10,000
ANSWER
The restrictions are
EXPLANATION
We were given the rational function,
The function is defined for all values of a, except
This has become a quadratic trinomial, so we need to split the middle term.
We do that by multiplying the coefficient of which is 5 by the constant term which is 3. This gives us 15.
The factors of 15 that adds up to 16 are 1 and 15.
We use these factors to split the middle term.
We now factor to get,
We factor further to get,
This implies that,
This gives
These are the restrictions.