At what constant, continuous annual rate should you deposit money into an account if you want to have $1,000,000 in 25 years? The account earns 5% interest, compounded continuously. Round to the nearest dollar.

FV = Pe^Yr

where FV = future value = 1,000,000
r here = .05
Y = 25
1,000,000 = P e^(1.25)
P = 1,000,000 / 3.49
P = 286,533

online calculator:

http://www.moneychimp.com/articles/finworks/continuous_compounding.htm

dP/dt = r P

dP/P = r dt

ln P = r t

e^ln P = e^(rt) + C

P = C e^(rt)

when t = 0, e^(rt) = 1
so C = value of P when t = 0
so
P = Po e^(rt)

How do I find the continuous rate though?

Oh, sorry

Try this, sinking fund "Continuous compounding at nominal rate r, uniform series"

http://ece.uprm.edu/~s016965/ININ%204015%20-%20Analisis%20Economico%20Para%20Ingenieros/Engineering%20Economic%20Analysis%208th%20ED.pdf

To find the constant, continuous annual rate at which you should deposit money, we need to use the formula for compound interest with continuous compounding:

A = P * e^(rt)

Where:
A = the future amount ($1,000,000)
P = the initial principal (unknown)
r = the annual interest rate (5% or 0.05)
t = the time in years (25)
e = Euler's number (approximately 2.71828)

We need to solve for P in this equation.

Dividing both sides of the equation by e^(rt), we get:

A / e^(rt) = P

Substituting the given values, we have:

$1,000,000 / e^(0.05*25) = P

Now, we can calculate this using a scientific calculator or an online tool.

Solving this equation gives us:

P ≈ $163,756.12

So, you would need to deposit approximately $163,756.12 in the account initially to have $1,000,000 in 25 years, with continuous compounding.

To find the constant, continuous annual rate at which you should deposit money, we divide P by the number of years:

$163,756.12 / 25 ≈ $6,550.25

Rounding to the nearest dollar, you should deposit approximately $6,550 per year at a continuous annual rate to reach $1,000,000 in 25 years, with continuous compounding.