(1 point) At what constant, continuous rate must money be deposited into an account if the account is to contain $22500 in 5 years? The account earns 6% interest compounded continuously.
Huh? You just said it earns 6%
I guess you want the amount invested. That would mean
Pe^(5*0.06) = 22500
solve for P
thats what I have been doing and it keeps telling me its wrong
If it is deposited at a continuous rate k it is
dA/dt = r A + k
not just dA/dt = r A
If it is deposited at a continuous rate k it is
dA/dt = r A + k
not just dA/dt = r A
messy!
In fact it started out zero
as oobleck suggested,
p e^(5(.06)) = 22500
p = 22500/e^.3 = 16,668.41
It must accept that answer, or else you typed it incorrectly.
but it says that the money is deposited at a continuous rate, not just the usual put it in and let it compound continuosly problem.
To determine the constant, continuous rate at which money must be deposited into the account, we can use the formula for compound interest in continuous compounding:
A = P * e^(rt)
Where:
A = the final amount in the account ($22500)
P = the initial principal (unknown)
e = Euler's number, approximately 2.71828
r = the interest rate (6% or 0.06)
t = the time period in years (5)
First, let's rearrange the formula to solve for the initial principal, P:
P = A / e^(rt)
Now we can substitute the known values into the formula:
P = 22500 / e^(0.06 * 5)
Calculating e^(0.06 * 5) yields approximately 1.34048. Plugging this value into the formula:
P = 22500 / 1.34048
P ≈ 16775.98
Therefore, the initial principal (or the amount that needs to be deposited into the account) is approximately $16775.98.
Now, let's calculate the constant, continuous rate at which money must be deposited. This rate is the ratio of the initial principal to the time period:
Rate = P / t
Rate = 16775.98 / 5
Rate ≈ 3355.20
Hence, money must be deposited into the account at a constant, continuous rate of approximately $3355.20 per year in order for the account to contain $22500 in 5 years.