SOLUTION: The size of a certain insect population at time t (in days) obeys the function P(t)=281e^0.09t 1. When will the population reach 1120 insects? Round your answer to the neares

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Question 1100500: The size of a certain insect population at time t (in days) obeys the function
P(t)=281e^0.09t
1. When will the population reach 1120 insects?
Round your answer to the nearest whole number.
2.When will the insect population double?
Round your answer to the nearest whole number.

Found 2 solutions by Fombitz, Theo:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
281e%5E%280.09t%29=1120
e%5E%280.09t%29=1120%2F281
0.09t=ln%281120%2F281%29
t=ln%281120%2F281%29%2F0.09
Work that out and round to the nearest whole number.
.
.
.
2%2A281=281e%5E%280.09t%29
e%5E%280.09t%29=2
0.09t=ln%282%29
t=ln%282%29%2F0.09
Work that out and round to the nearest whole number.

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
p(t) = 281 * e^(.09 * t)

t is the time in days.

the population will reach 1120 insects when p(t) = 1120.

formula becomes 1120 = 281 * e^(.09 * t)

divide both sides by 281 to get:
1120/281 = e^(.09 * t)
take the natural log of both sides of this equaiton to get:
ln(1120/281) = ln(e^(.09 * t))
this becomes:
ln(1120/281) = .09 * t * ln(e) which becomes:
ln(1120/281) = .09 * t
solve for t to get:
t = ln(1120/281) / .09 = 15.36365883.

the population will reach 1120 in 15.36365883 days.

281 * e^(.09 * 15.36365883) = 1120, confirming the solution is correct.

2 basic principles involved.

ln(e^x) = x*ln(e).
ln(e) = 1

to find out when the insect population will double, do the following.

p(t) = 281 * e^(.09 * t)

let p(t) = 2 * 281 = 562

formula becomes 562 = 281 * e^(.09 * t)

do the following:

divide both sides of the equation by 281 to get:
562/281 = e^(.09 * t)
simplify to get:
2 = e^(.09 * t)
take the natural log of both sides of the equation to get:
ln(2) = ln(e^(.09 * t))
this becomes:
ln(2) = .09 * t * ln(e) which becomes:
ln(2) = .09 * t
solve for t to get:
t = ln(2) / .09 = 7.70163534

the insect population will double in 7.70163534 days.

281 * e^(.09 * 7.70163534) = 562, confirming the solution is correct, because 562 is double 281.

rounding your solutions to the nearest integer results in.

population will reach 1120 in 15 years.
population will double in 8 years.