Answer:
10.4 hours
Step-by-step explanation:
It is convenient to write the exponential growth function in these terms:
population = (initial value)·((value at later time)/(initial value))^exponent
where ...
exponent = (t - (initial time))/(time difference to later time)
Here, we have ...
- initial value = 456
- initial time = 2 . . . hours
- value at later time = 988
- time difference to later time = 4.2 -2 = 2.2 . . . hours
So, the exponential function is ...
p(t) = 456·(988/456)^((t-2)/2.2)
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We can find t for p(t) = 8700 a couple of ways. My favorite is to use a graphing calculator to find t such that ...
p(t) -8700 = 0
This shows t ≈ 10.4 hours.
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Alternatively, you can solve the equation using algebra.
8700 = 456·(988/456)^((t-2)/2.2)
8700/456 = (988/456)^((t-2)/2.2) . . . . . . . . . . .divide by 456
log(8700/456) = (t -2)/2.2 · log(988/456) . . . . taking logs
2.2·log(8700/456)/log(988/456) = t -2 . . . . . . multiply by the inverse of the coefficient of t
t = 2 + 2.2(log(8700/456)/log(988/456)) ≈ 10.389774
We can expect the population to be 8700 at 10.4 hours.
