



Sometimes
how fast something grows



depends
on how much we have. 





When
bacteria grows, 


the
amount of new bacteria you get every minute 


depends
on how much you already have. 


The
way this looks in an equation is: 





Amount
You Get = Amount You Start With × e^{constant × time}






That
is, the amount that you have after some time has passed 


is
equal to the amount you started with 


times
"e" raised to a power made up of 


the
amount of time that has passed times some constant (a number) 


The
constant is there to make sure the units work out right. 


Usually
in the equation: 


We
use the letter k for the constant (this will be a different 


We
use the letter t for the time 


We
use the letter Q for the quantity of stuff we have at the end 


We
use the letters Q_{o} for the original quantity of stuff we
start with. 





So
put it all together, the equation look like this: 





Q = Q_{o}e^{kt}






Example: 


The
rate that mystery bacteria Q grows depends on the amount present. 


After
3 minutes, Q has grown from 100 to 500 units 


How
much is there after 10 minutes? 





First
we need to find out what k is. 


k
will be different for each problem 


so
we have to find k for each problem we do. 





Set
up the problem. 


The
amount we start with (Qo) is 100 


The
amount we end up with (Q) is 500 


The
time (t) is 3 minutes 


e
is 2.718 ... 





That
means we have: 


Q = Q_{o}e^{kt}



500 = 100e^{k3} 





Now
we solve this problem for k. 


First
we get rid of the big numbers ... 








Since
k is an exponent we need our special log trick 


to
turn it into just a factor. 


Take
the log of each side. 


It's
easier in this case to use the ln type of log ... 





ln 5 = ln
e^{3k}






Now
use the log trick to bring down the exponents ... 





ln 5 = 3k
ln e






Now
remember what ln e means 


It
means "the power that e must be raised to, to get e" 


That
gives us: 


ln
5 = 3k






Now
divide each side by 3 ... 








ln
5 is just a number, so we can solve this one ... 











Now
we can put .536 in for k and solve the rest of the problem. 


Remember
the rest of the problem? 


How
much is there after 10 minutes? 


That
means: 





Set
up the problem. 


The
amount we start with (Qo) is 100 


The
amount we end up with (Q) is THE UNKNOWN 


The
time (t) is 10 minutes 


The
constant (k) is .536 





So
we have: 


Q = Q_{o}e^{kt}



Q = 100e^{10
× .536}






Simplify
this a bit: 


Q = 100e^{5.36}






Now
we need a calculator. 


On
my calculator, this is how you punch this one in ... 











So
after 10 minutes we have over 21,000 units of bacteria Q 





Example: 


The
rate that the population of "Our Town" grows 


depends
on the number of people in the town. 


In
1900 there were 1000 people. 


In
1950 there were 2000 people. 


When
will there be 10,000 people in "Our Town"? 





Since
we're looking for total people, 


let's
use P instead of Q. 





So
we have: 



P_{o}
= 1000 



P =
2000 



t = 50 (from
1900 to 1950) 



k = the
first thing to solve for ... 





P = P_{o}e^{kt } 


2000
= 1000e^{50k}






Divide
each side by 1000 ... 








Now
take the natural log (ln) of each side ... 





ln
2 = ln e^{50k}






and
use the log trick (don't forget, ln e = 1) 







ln 2 =
50k (ln 2 =
.693) 




.693 = 50k 





Divide
both sides by 50 to get k by itself ... 











That's
great. Now we have k. 


But
we're not done yet. 


The
problem asked: 


'When
will there be 10,000 people in "Our Town"?' 


Finding
out WHEN means we need to solve for t. 


We
have: 



P_{o}
= 1000 



P =
10,000 



k = .0139 



t = WE
DON'T KNOW 





Let's
do it! 


P = P_{o}e^{kt } 


10,000 = 1,000e^{.0139t } 





Divide
both sides by 1000 ... 











Now
we need our log trick (OK, our ln trick) 











We
need to solve for t so divide both sides by .0139 ... 











So
it takes 165.7 years for the population of "Our Town" 


to
grow from 1000 to 10,000 people. 


But
the question was WHEN will there be 10,000 people 


in
Our Town. 


That
means WHAT YEAR. 


We
started with 1000 people in 1900. 


165.7
years later we will have 10,000 people. 


The
year will be 1900 + 165.7 = 2065.7. 





So
Our Town will have 10,000 people in about the year 2065 or 2066 





When
stuff grows like this it is called EXPONENTIAL GROWTH. 





copyright 2005 Bruce Kirkpatrick 
