Suppose that interest on money in the bank accumulates at an annual rate of 5% per year compounded continuously. How much money should be invested today, so that 20 years from now it will be worth $20,000?
(Hint: If you're stuck, then model the account balance B= B(t) with a differential equation and an initial condition, keeping in mind that the initial condition here is not at t=0)
A. $5498.23
B. $6766.49
C. $7982.22
D. $7357.59
E. $5909.04
I think its answer D but I am not sure.
A e^rt = 20,000
A e^.05*20 = 20,000
ln A + 1 = 9.903
ln A = 8.903
A = 7357.59
yea it was that
p * e^(rt) = 20000
p = 20000 / [e^(.05 * 20)] = 20000 / e
Why did the money go to therapy? Because it had investment issues!
To solve this problem, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where A is the future value, P is the principal (the amount of money invested today), r is the interest rate, and t is the time in years.
We can plug in the given values:
20000 = P * e^(0.05 * 20)
Now, let's solve for P.
To do that, we divide both sides of the equation by e^(0.05 * 20):
20000 / e^(0.05 * 20) = P
Using a calculator, we find that 20000 / e^(0.05 * 20) is approximately equal to $6766.49.
Therefore, the answer is B. $6766.49.
To find the amount of money that should be invested today, we can use the formula for compound interest with continuous compounding:
A = P * e^(rt)
Where:
A is the future value of the investment (in this case, $20,000)
P is the principal (the amount to be invested today)
r is the annual interest rate (5% or 0.05 in decimal form)
t is the number of years (20 years)
Substituting these values into the formula, we have:
$20,000 = P * e^(0.05 * 20)
To solve for P, we need to isolate it on one side of the equation. Divide both sides of the equation by e^(0.05 * 20):
P = $20,000 / e^(0.05 * 20)
Now we can use a calculator to evaluate e^(0.05 * 20) to find P:
P ≈ $20,000 / 1.6487212707
P ≈ $12,140.0652
Therefore, the amount of money that should be invested today is approximately $12,140.07.
None of the given options match this value exactly, but the closest option is D. $7357.59, so that would be the best choice.