Suppose an investment earns 5.1 % interest compounded continuously. How long will it take an investment of $2000 to be worth $3000? Round your answer to the nearest hundredth.

Answer 1

To solve this problem, we can use the formula for continuously compounded interest:

A = P * e^(rt)

Where:
A = the amount of money in the account after t years
P = the principal amount ($2000 in this case)
r = the annual interest rate (5.1% or 0.051 in decimal form)
t = the time in years
e = Euler's number, approximately 2.71828

We want to find out how long it will take for the investment to be worth $3000, so we set A = $3000 and solve for t:

$3000 = $2000 * e^(0.051t)

Dividing both sides by $2000:

1.5 = e^(0.051t)

Taking the natural logarithm of both sides:

ln(1.5) = 0.051t

Solving for t:

t = ln(1.5) / 0.051
t ≈ 10.24

So, it will take approximately 10.24 years for the investment to be worth $3000.