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. Calculate to the nearest hundredth of a year how long it takes for an amount of money to double if interest is compounded continuously at 6.2%. Round to the nearest tenth.

Question

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. Calculate to the nearest hundredth of a year how long it takes for an amount of money to double if interest is compounded continuously at 6.2%. Round to the nearest tenth.

Answer 1

This means that

A/P=2, or
e^(rt)=2
log(e^(rt))=log_e2
rt=log_e2
t=(log_e2)/r
=(log_e2)/0.062 years
=?

Answer 2

This means that

A/P=2, or
e^(rt)=2
log(e^(rt))=log_e2
rt=log_e2
t=(log_e2)/r
=(log_e2)/0.062 years
=?

Note:
This is actually the rule of 69.31, as follows:
The number of years to double money multiplied by the interest rate of p% compounded continuously is 69.31.

Answer 3

Eli deposited $1400 at 6.5% interest compounded quarterly. How much money will he have at the end of 8 years?

Answer 4

An initial amount of money is placed in an account at an interest rate of per year, compounded continuously. After three years, there is in the account. Find the initial amount placed in the account. Round your answer to the nearest cent.

Answer 5

0.3=10^06x

Answer 6

To calculate how long it takes for an amount of money to double with continuously compounded interest, we need to find the time 't' when the amount 'A' becomes twice the principal 'P'.

Let's go step by step:

1. First, we need to determine the value of 'r', the annual interest rate, which is given as 6.2% or 0.062 in decimal form.

2. Next, we need to find the value of 'A' when it is double the principal 'P'. Since we want to find the time it takes for the amount to double, let A = 2P.

3. Now, we substitute the given values into the continuous compound interest formula:

2P = Pe^(rt)

4. We can simplify the equation by canceling out 'P' from both sides:

2 = e^(rt)

5. Taking the natural logarithm (ln) of both sides:

ln(2) = ln(e^(rt))

ln(2) = rt(ln(e))

6. Since ln(e) is equal to 1, the equation becomes:

ln(2) = rt

7. We can solve for 't' by isolating it:

t = ln(2)/r

8. Now we substitute the value of 'r' (0.062) into the formula:

t = ln(2)/0.062

9. Using a calculator, calculate the value of ln(2) and then divide it by 0.062.

t ≈ 11.18

10. Round the value to the nearest hundredth of a year, which gives:

t ≈ 11.2 years (rounded to the nearest tenth)

Therefore, it takes approximately 11.2 years for the amount of money to double with continuously compounded interest at a rate of 6.2%.