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 3.8%. Round to the nearest tenth.
so you are solving
2 = 1(e^(.038t))
ln2 = ln(e^(.038t))
ln2 = .038t(lne) but lne = 1
.038t = ln2
t = ln2/.038 = 18.24 yrs
Suppose you invest $2500 at an annual interest rate of 3% compounded continuously. How much will you have in the account after 7 years? Round the solution to the nearest dollar. ?
To determine how long it takes for an amount of money to double with continuous compounding, we need to solve the equation A = 2P, where A is the new amount, P is the principal, r is the interest rate, and t is the time in years.
In this case, we know that the interest rate is 3.8% or 0.038, and we want to find the value of t. Let's set up the equation:
2P = Pe^(0.038t)
Next, we can cancel out the common factor of P:
2 = e^(0.038t)
To calculate t, we need to take the natural logarithm (ln) of both sides to remove the exponential term:
ln(2) = ln(e^(0.038t))
Using the property that ln(e^x) equals x, we simplify the equation:
ln(2) = 0.038t
Now, divide both sides by 0.038:
t = ln(2) / 0.038
Using a scientific calculator or an online calculator, the value for t is approximately 18.24 years (rounded to two decimal places).
Therefore, it takes approximately 18.24 years (rounded to the nearest hundredth of a year) for the amount of money to double if interest is compounded continuously at a rate of 3.8%.