SOLUTION: The amount of money in an account with continuously compounded interest is given by the formula A = Pert, where P is the principal, r is the annual interest rate, and t is the time

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Question 1194673: The amount of money in an account with continuously compounded interest is given by the formula A = Pert, where P is the principal, r is the annual interest rate, and t is the time in years.
Calculate to the nearest tenth of a year how long it takes for an amount of money to double if interest is compounded continuously at 7.5%.
Found 2 solutions by ikleyn, MathTherapy:
Answer by ikleyn(52754) About Me  (Show Source):
You can put this solution on YOUR website!
.
The amount of money in an account with continuously compounded interest
is given by the formula A = Pert, where P is the principal, r is the annual interest rate,
and t is the time in years.
Calculate to the nearest tenth of a year how long it takes for an amount of money
to double if interest is compounded continuously at 7.5%.
~~~~~~~~~~~~~~~~~~~~~~~ 

Starting equation is

    2P = P%2Ae%5E%280.075%2At%29.


Divide both sides by P
    
   2  = e%5E%280.075%2At%29.


Take natural logarithm of both sides

    ln(2) = 0.075*t.


Express t

    t = ln%282%29%2F0.075 = 9.24.


ANSWER.  9.2 years, rounded.

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.


Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this online textbook under the topic "Logarithms".


Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.


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The post-solution note:


        I entirely agree with the note by tutor  @MathTherapy . . .



Answer by MathTherapy(10549) About Me  (Show Source):
You can put this solution on YOUR website!

The amount of money in an account with continuously compounded interest is given by the formula A = Pert, where P is the principal, r is the annual interest rate, and t is the time in years.
Calculate to the nearest tenth of a year how long it takes for an amount of money to double if interest is compounded continuously at 7.5%.
Approximate number of years: 9.241962407 ≈ 9.2 years (to 1 decimal place, as requested). 

The above answer is more than likely the one that's being sought, but in reality, in 9.2 years, the invested amount will NOT double. It will get close to 

doubling, but it NEVER quite doubles. A better answer would be 9.3 (to 1 decimal place) years, since in that period of time, the amount will SURELY double!