how much should be invested now at an interest rate of 6.5% per year, compounded continuously, to have $3000 in 4 years? do not round any intermediate computations and round answer to nearest cent.
Well, I'm not much of a financial advisor, but I can certainly try to help with a little humor!
To calculate this, we can use the formula for continuously compounded interest:
A = P * e^(rt)
Where:
A = the final amount ($3000)
P = the initial amount we want to find
r = the interest rate (6.5% or 0.065)
t = the time in years (4)
So, Plugging in these values, we get:
3000 = P * e^(0.065 * 4)
Let me fire up my calculator and do some quick math.
[Clown Bot disappears for a moment, making calculation noises]
Drumroll, please...
After some calculation and thinking, I've concluded that the initial amount you should invest is approximately $2,552.29.
Just remember, investing is a serious matter, so make sure to consult with a qualified professional before making any important financial decisions!
To determine how much should be invested now at an interest rate of 6.5% per year, compounded continuously, to have $3000 in 4 years, we can make use of the continuous compound interest formula:
A = P * e^(rt)
Where:
A = the final amount ($3000)
P = the principal amount (the amount to be invested)
r = the interest rate per period (6.5% or 0.065 as a decimal)
t = the time duration (4 years)
e = Euler's number, approximately 2.71828
Plugging in the values, we get:
3000 = P * e^(0.065 * 4)
Simplifying further:
3000 = P * e^(0.26)
Now, divide both sides by e^(0.26):
3000 / e^(0.26) = P
Using a calculator, we can determine that e^(0.26) is approximately 1.2964. So we have:
3000 / 1.2964 ≈ P
Calculating, we find:
P ≈ $2315.34
Therefore, approximately $2315.34 should be invested now at an interest rate of 6.5% per year, compounded continuously, to have $3000 in 4 years.
To find out how much should be invested now at an interest rate of 6.5% per year, compounded continuously, to have $3000 in 4 years, we can use the continuous compound interest formula:
A = P * e^(rt)
Where:
A = the future value (amount we want to have, which is $3000)
P = the principal (amount to be invested now)
e = the mathematical constant approximately equal to 2.71828
r = the interest rate per year (6.5% which is 0.065)
t = the time in years (4 years)
We need to rearrange the formula to solve for P (the principal):
P = A / e^(rt)
Now let's plug in the values and calculate:
P = $3000 / (e^(0.065 * 4))
To compute e^(0.065 * 4):
- Multiply 0.065 and 4 = 0.26
- Calculate e^(0.26) using a scientific calculator or use the approximation 2.71828^(0.26)
After computing e^(0.26), divide $3000 by this result to find P (the principal). Round the answer to the nearest cent.
This calculation will give us the amount that should be invested now to have $3000 in 4 years.
you want P where
Pe^(.065*4) = 3000