SOLUTION: An insect colony grow exponentially from 150 to 3000 in 4 months. How long will it take for the colony to reach 50,000? Use the formula A(m)=Pe^rt.(where time is in months)

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Question 881792: An insect colony grow exponentially from 150 to 3000 in 4 months. How long will it take for the colony to reach 50,000? Use the formula A(m)=Pe^rt.(where time is in months)
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Write it this way: A%28t%29=Pe%5E%28rt%29 and know that t is number of MONTHS.

You first want to find the rate, r. Do this symbolically for substitution convenience.

Drop the notation for the t in the function name also for convenience.
ln%28A%29=ln%28Pe%5E%28rt%29%29
rt%2Aln%28P%29=ln%28A%29
r=%281%2Ft%29%28ln%28A%29%2Fln%28P%29%29
NOW substitute.
r=%281%2F4%29%28ln%283000%29%2Fln%28150%29%29
r=0.3994 maybe, but better as highlight_green%28r=0.399%29.

The question is find t for A(t)=50000.
Remember the step saying, rt%2Aln%28P%29=ln%28A%29 ? Now, just divide left and right by r to solve symbolically for t:
t=%281%2Fr%29%28ln%28A%29%2Fln%28P%29%29
and substitute the known and found values.
t=%281%2F0.3994%29%28ln%2850000%29%2Fln%28150%29%29, which means time starting from the initial 150 colony.
highlight%28t=5.4%29 months.