Question

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%.

Answer 1

For one thing that formula A=Pert could be written better than that. (See the attached formula for "Total".)
For the question you asked, you will need the other attached formula that says "Years".
Let's say we want to calculate when $100 becomes $200
Years = natural log (Total/Principal) / rate
Years = natural log (200/100) / .075
Years = natural log (2) / .075
Years = 0.69314718056 / .075
Years = 9.2419624075

Answer 2

9 years

Explanation:

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

Initial amount invested is P and when the amount of money doubles then A becomes 2P

A= 2P

P=P

r= 7.5%= 0.075

t=t

Now we use formula A=Pe^rt

`2P= P e^(0.075*t)`

Divide by P on both sides

`2=e^(0.075*t)`

To solve for t, take ln on both sides'

`ln(2)=ln(e^(0.075*t))`

`ln(2)=(0.075*t)ln(e)`

ln(2)= 0.075 *t

Divide by 0.075 on both sides

t= 9.24196