SOLUTION: The size of certain insect poplation is given by P(t) =2000 e^(0.01t), where t is measured in days. a) How many insects were present initially? b) what is the growth rate

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Question 1157354: The size of certain insect poplation is given by
P(t) =2000 e^(0.01t), where t is measured in days.
a) How many insects were present initially?
b) what is the growth rate?
c) At what time will the population double?
d) give a differential equation satisfied by p(t)?
d) how fast the becteria is growing when it reaches 10.000?
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
The size of certain insect population is given by
P%28t%29+=2000+e%5E%280.01t%29, where t is measured in days.
a) How many insects were present initially?
the initial amount is P%280%29=2000+
b) what is the growth rate?
The growth rate is 0.01
c) At what time will the population double?
double amount is P%28t%29+=4000
4000+=2000+e%5E%280.01t%29
4000%2F2000+=e%5E%280.01t%29
e%5E%280.01t%29=2......take natural log of both sides
ln%28e%5E%280.01t%29%29=ln%282%29
%280.01t%29ln%28e%29=ln%282%29
0.01t=ln%282%29%2Fln%28e%29.........ln%28e%29=1
0.01t=0.69314718
t=0.69314718%2F0.01
t≈69.315+days

d) give a differential equation satisfied by p%28t%29?
take the constant out : %28a%2A+f+%29'=a* f'
P'%28t%29+=2000+%2A%28d%2Fdt%29%28e%5E%280.01t%29%29
apply the chain rule:%28df%28u%29%2Fdt%29=%28df%2Fdu%29%2A%28du%2Fdx%29
f=e%5Eu, u=0.01t
P'%28t%29+=2000+%2A%28d%2Fdu%29%28e%5Eu%29%2A%28d%2Fdt%29%280.01t%29
%28d%2Fdu%29%28e%5Eu%29=e%5Eu
%28d%2Fdt%29%280.01t%29=0.01
substitute u back

P'%28t%29=2000%2Ae%5E%280.01t%29%2A0.01

P'%28t%29+=+20%2Ae%5E%280.01t%29

d) how fast the bacteria is growing when it reaches 10000?
10000+=2000%2Ae%5E%280.01t%29
10000+%2F2000+=e%5E%280.01t%29
5+=e%5E%280.01t%29
ln%28e%5E%280.01t%29%29=ln%285%29
%280.01t%29ln%28e%29=ln%285%29
0.01t=ln%285%29
t=ln%285%29%2F0.01
t≈160.944+days