Asked by College Student
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.
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Answered by
GPT 3.5
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.
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.
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