Question 731574: Suppose you deposit $500 in an account with an annual interest rate of 8% compounded monthly.
a. Find an equation that gives the amount of money in the account after t years.
b. Find the amount of money in the account after 5 years.
c. How many years will it take for the account to contain $1000?
d. If the interest were compounded continuously, how much money would the account contain after 5 years?
Thank You So Much!
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! Interest compounded monthly:
The interest for 1 month would be  of 8%, meaning  of the initial balance that month.
So at the end of the first month you would have a balance (in $) of
So the new balance would be the initial balance times  .
During the second month, you would be earning interest on the whole of that $ and that balance would be multiplied times to get the new balance of
After  months you would have a total of
$
a. At the end of 5 years you would have accumulated interest during  and would have a total of
$ = $ (rounded)
b. To get to  you will need months, and we can find like this:
From
taking logarithms on both sides, we get
 -->  --> 
That calculates as about  (rounded)
If you need $1000, you will have to wait for 105 months because
after 104 months you will have
$ = $ (rounded)
but after 105 months you will have
$ = $ (rounded)
c. Interest compounded monthly:
If the interest was calculated and added to the balance after shorter and shorter periods, you would gain a little bit more as the periods were shortened, getting as close as you want, but never going over a certain limit.
That limit is what they call continuous compounding.
The balance with continuous compounding is given by the function
 = $ with  = number of years.
In that expression,
$ is the initial deposit,
 is the 8% interest rate expresed as a decimal,
and  is an irrational number, like  .
For  we get the balance after 5 years as
$ = $ = $
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