SOLUTION: Mia invests $2,000 in a money market account that earns 5% annual interest, compounded continuously. Approximately how many years will it take her money to grow to the $4,500 she n

Algebra ->  Numeric Fractions Calculators, Lesson and Practice -> SOLUTION: Mia invests $2,000 in a money market account that earns 5% annual interest, compounded continuously. Approximately how many years will it take her money to grow to the $4,500 she n      Log On


   

Question 1205115: Mia invests $2,000 in a money market account that earns 5% annual interest, compounded continuously. Approximately how many years will it take her money to grow to the $4,500 she needs for her small business start-up? (Round your answer to one decimal place.)
Found 2 solutions by ikleyn, MathLover1:
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
Mia invests $2,000 in a money market account that earns 5% annual interest, compounded continuously.
Approximately how many years will it take her money to grow to the $4,500 she needs for her small
business start-up? (Round your answer to one decimal place.)
~~~~~~~~~~~~~~~~~~~~~~~ 

Using the formula for continuously compounded account, write

    4500 = 2000%2Ae%5E%280.05%2At%29.


Divide both sides by 2000

    4500%2F2000 = e%5E%280.05%2At%29.


It is the same as

    2.25 = e%5E%280.05%2At%29.


Take natural logarithm of both sides

    ln(2.25) = 0.05*t,

    t = ln%282.25%29%2F0.05 = 16.22


Round to one decimal place

    t = 16.2 years.    ANSWER

Solved.

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To see many other similar  (and different)  solved problems on continuously compounded accounts,  look into the lesson
    - Problems on continuously compound accounts
in this site.

After reading this lesson,  you will tackle such problems on your own without asking for help from outside.



Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

This formula says, when an amount+P is invested for the time 't' with the interest rate is r% compounded continuously, then the final amount is,
A+=+P+%2Ae%5E%28rt%29

The initial amount is P+=+2000.
The interest rate is, r+=+5% = 5%2F100+=+0.05
The amount after t+years =4500
Time is, t=?

4500+=+2000%2A+e%5E%280.05t%29
4500%2F2000=e%5E%280.05t%29
e%5E%280.05t%29=9%2F4....take natural log of both sides
ln%28e%5E%280.05t%29%29=ln%289%2F4%29
%280.05t%29ln%28e%29=ln%289%29-ln%284%29...ln%28e%29=1
0.05t=2.19722-1.386294361
0.05t=0.810925639
t=0.810925639%2F0.05
t=16.2
it will take her 16.2 years for money to grow to the $4500+