Find the values of the constants C and b so that the curve y = Cb^(x) contains the points (4 , 3) and (5 , 6) .

for the point (4,3) we can write 
for the point (5,6) we can write 

then taking the quotient of these two equations leads to


s we see that



Now we can use either point (we'll go with (4,3)) and b directly to write





so,

If $1000 is invested in an account paying 12% annually compounded continously, how long will it take to double the investment?

For continuous compounding we use:


so we can write:



taking the natural log of both sides leads to



so time is about 5.78 years

how long will it take for the population of a certain country to quadruple if its annual growth rate is 2.5% round to the nearest year.

Growth model: 
 
which we simplify to get 

 
solving for t ... 
 


 
this is about 81 years

The population P of a city if given by P=2500e^(kt) where t is time of the years , with t=0 corresponding to the years 1980 .In 1935 ,the population was 3350 .Find the value of k and use this result to predict the population in the years 2000

t=1935-1980= -45 


3350=2500e^(k*(-45)) 

e^(k*(-45))=3350/2500 


ln(e^(k*(-45)))=ln(3350/2500) 

k=ln(3350/2500)/(-45)=-0.0065 


In 2000 t=20, so... 


P=2500e^(-0.0065*20)= 2196 

:)

Hand sanitizer kills a certain bacterium 80% of the time on each application. What is the minimum number of times you must use the hand sanitizer to be 98% certain you have killed that bacterium?

We'll write: 
 
and 98% implies P=1-0.98 = 0.02 

Solving... 
 

 

so we need to apply 18 times 

:)

Exponential growth

Given: There are 7.3 billion, that is, 7.3*10^9
people on Earth. If a flu virus starts
with one person and within 1 day four more
people are infected, then each of those four
people in 1 day infects 4 people each,
and so on, how many days before everyone
on earth is infected?

Solution:

This is an exponential growth problem where the population quadruples every day. Our model equation is:


with
, 
which leads to...





t is about...16.4 days