http://www.ck12.org Chapter 6. Transcendental Functions
t=ln 2k
= 0 ln 2. 409
t= 1 .7 hours.This tells us that after about 1.7 hours (around 100 minutes) the population of the bacteria will double in number.
Compound Interest
Investors and bankers depend on compound interest to increase their investment. Traditionally, banks added interest
after certain periods of time, such as a month or a year, and the phrase was “the interest is being compounded
monthly or yearly.” With the advent of computers, the compunding could be done daily or even more often. Our
exponential model represents continuous, or instantaneous, compounding, and it is a good model of current banking
practices. Our model states that
A=Pert,wherePis the initial investment (present value) andAis the future value of the investment after timetat an interest
rate ofr.The interest rateris usually given in percentage per year. The rate must be converted to a decimal number,
andtmust be expressed in years. The example below illustrates this model.
Example 3:
An investor invests an amount of $10,000 and discovers that its value has doubled in 5 years. What is the annual
interest rate that this investment is earning?
Solution:
We use the exponential growth model for continuously compounded interest,
A=Pert
20 , 000 = 10 , 000 er(^5 )
2 =e^5 r
ln 2= 5 r.Thus
r=ln 2 5
= 0. 139
r= 13 .9%The investment has grown at a rate of 13.9% per year.
Example 4:
Going back to the previous example, how long will it take the invested money to triple?
Solution: